A Dynamic Survey of Graph Labeling

Joseph A. Gallian

Department of Mathematics and Statistics University of Minnesota Duluth Duluth, Minnesota 55812

                          jgallian@d.umn.edu

Submitted: September 1, 1996; Accepted: November 14, 1997 Twelfth edition published: January 31, 2009

Mathematics Subject Classifications: 05C78

Abstract

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

Contents

1 Introduction 5

2 Graceful and Harmonious Labelings 7 2.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Cycle-Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Product Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Disconnected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Joins of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Table 1: Summary of Graceful Results . . . . . . . . . . . . . . . . . . . . 24 Table 2: Summary of Harmonious Results . . . . . . . . . . . . . . . . . . 28

3 Variations of Graceful Labelings 31

3.1 α-labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Table 3: Summary of Results on α-labelings . . . . . . . . . . . . . . . . . 37 3.2 k-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 γ-Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Skolem-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 Odd-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 Graceful-like Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8 The Friendly Index–Balance Index . . . . . . . . . . . . . . . . . . . . . . 55 3.9 k-equitable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.10 Hamming-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Variations of Harmonious Labelings 59 4.1 Sequential and Strongly c-harmonious Labelings . . . . . . . . . . . . . . . 59 4.2 (k, d)-arithmetic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 (k, d)-indexable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Elegant Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Felicitous Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Magic-type Labelings 67

5.1 Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Table 4: Summary of Magic Labelings . . . . . . . . . . . . . . . . . . . . 72 5.2 Edge-magic Total and Super Edge-magic Labelings . . . . . . . . . . . . . 74

Table 5: Summary of Edge-magic Total Labelings . . . . . . . . . . . . . . 82 Table 6: Summary of Super Edge-magic Labelings . . . . . . . . . . . . . . 85

5.3 Vertex-magic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 89 Table 7: Summary of Vertex-magic Total Labelings . . . . . . . . . . . . . 95 Table 8: Summary of Super Vertex-magic Total Labelings . . . . . . . . . 97 Table 9: Summary of Totally Magic Labelings . . . . . . . . . . . . . . . . 98

5.4 Magic Labelings of Type (a, b, c) . . . . . . . . . . . . . . . . . . . . . . . . 99

Table 10: Summary of Magic Labelings of Type (a, b, c) . . . . . . . . . . . 101 5.5 Other Types of Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Antimagic-type Labelings 105

6.1 Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Table 11: Summary of Antimagic Labelings . . . . . . . . . . . . . . . . . 109 Table 12: Summary of (a, d)-Edge-Antimagic Vertex Labelings . . . . . . . 110 Table 13: Summary of (a, d)-Antimagic Labelings . . . . . . . . . . . . . . 111

6.2 (a, d)-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . 112

Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)-

Vertex-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . 119 Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings . . . . . . . . 120 Table 16: Summary of (a, d)-Super-Edge-Antimagic Total Labelings …. 122

6.3 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) …. 123 Table 17: Summary of Face Antimagic Labelings . . . . . . . . . . . . . . . 126 Table 18: Summary of d-antimagic Labelings of Type (1,1,1) . . . . . . . . 127

6.4 Product Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Miscellaneous Labelings 129

7.1 Sum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Table 19: Summary of Sum Graph Labelings . . . . . . . . . . . . . . . . . 135 7.2 Prime and Vertex Prime Labelings . . . . . . . . . . . . . . . . . . . . . . 136

Table 20: Summary of Prime Labelings . . . . . . . . . . . . . . . . . . . . 139 Table 21: Summary of Vertex Prime Labelings . . . . . . . . . . . . . . . . 142

7.3 Edge-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Table 22: Summary of Edge-graceful Labelings . . . . . . . . . . . . . . . . 150 7.4 Radio Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Line-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6 Representations of Graphs modulo n . . . . . . . . . . . . . . . . . . . . . 153 7.7 k-sequential Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.8 IC-colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.9 Product Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.10 Prime Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.11 Geometric Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.12 Sequentially Additive Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.13 Strongly Multiplicative Graphs . . . . . . . . . . . . . . . . . . . . . . . . 157 7.14 Mean Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.15 Permutation and Combination Graphs . . . . . . . . . . . . . . . . . . . . 159 7.16 Strongly ⋆-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.17 Irregular total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.18 Sigma Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.19 Set Graceful and Set Sequential Graphs . . . . . . . . . . . . . . . . . . . . 161 7.20 Divisor Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.21 Difference Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

References 162

Index 213

1 Introduction

Most graph labeling methods trace their origin to one introduced by Rosa [733] in 1967, or one given by Graham and Sloane [380] in 1980. Rosa [733] called a function f a βvaluation of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, …, q} such that, when each edge xy is assigned the label |f(x) − f(y)|, the resulting edge labels are distinct. Golomb [371] subsequently called such labelings graceful and this is now the popular term. Rosa introduced β-valuations as well as a number of other labelings as tools for decomposing the complete graph into isomorphic subgraphs. In particular, β-valuations originated as a means of attacking the conjecture of Ringel [723] that K_{2n+1} can be decomposed into 2n + 1 subgraphs that are all isomorphic to a given tree with n edges. Although an unpublished result of Erdős says that most graphs are not graceful (cf. [380]), most graphs that have some sort of regularity of structure are graceful. Sheppard [807] has shown that there are exactly q! gracefully labeled graphs with q edges. Rosa [733] has identified essentially three reasons why a graph fails to be graceful: (1) G has “too many vertices” and “not enough edges,” (2) G “has too many edges,” and (3) G “has the wrong parity.” An infinite class of graphs that are not graceful for the second reason is given in [170]. As an example of the third condition Rosa [733] has shown that if every vertex has even degree and the number of edges is congruent to 1 or 2 (mod 4) then the graph is not graceful. In particular, the cycles C_{4n+1} and C_{4n+2} are not graceful.

Acharya [11] proved that every graph can be embedded as an induced subgraph of a graceful graph and a connected graph can be embedded as an induced subgraph of a graceful connected graph. Acharya, Rao, and Arumugam [26] proved: every trianglefree graph can be embedded as an induced subgraph of a triangle-free graceful graph; every planar graph can be embedded as an induced subgraph of a planar graceful graph; and every tree can be embedded as an induced subgraph of a graceful tree. These results demonstrate that there is no forbidden subgraph characterization of these particular kinds of graceful graphs.

Harmonious graphs naturally arose in the study by Graham and Sloane [380] of modular versions of additive bases problems stemming from error-correcting codes. They defined a graph G with q edges to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x) + f(y) (mod q), the resulting edge labels are distinct. When G is a tree, exactly one label may be used on two vertices. Analogous to the “parity” necessity condition for graceful graphs, Graham and Sloane proved that if a harmonious graph has an even number of edges q and the degree of every vertex is divisible by 2^{k} then q is divisible by 2^{k+1}. Thus, for example, a book with seven pages (i.e., the cartesian product of the complete bipartite graph K_{1,7} and a path of length 1) is not harmonious. Liu and Zhang [620] have generalized this condition as follows: if a harmonious graph with q edges has degree sequence d_{1}, d_{2}, …, d_{p} then gcd(d_{1}, d_{2}, … d_{p}, q) divides q(q − 1)/2. They have also proved that every graph is a subgraph of a harmonious graph. More generally, Sethuraman and Elumalai [779] have shown that any given set of graphs G_{1}, G_{2}, …, G_{t} can be embedded in a graceful or harmonious graph. Determining whether a graph has a harmonious labeling was shown to be NP-complete by Auparajita, Dulawat, and Rathore in 2001 (see [518]).

Over the past three decades in excess of 1000 papers have spawned a bewildering array of graph labeling methods. Despite the unabated procession of papers, there are few general results on graph labelings. Indeed, the papers focus on particular classes of graphs and methods, and feature ad hoc arguments. In part because many of the papers have appeared in journals not widely available, frequently the same classes of graphs have been done by several authors and in some cases the same terminology is used for different concepts. In this article, we survey what is known about numerous graph labeling methods. The author requests that he be sent preprints and reprints as well as corrections for inclusion in the updated versions of the survey.

Earlier surveys, restricted to one or two labeling methods, include [159], [175], [495], [346], and [348]. The book edited by Acharya, Arumugam, and Rosa [16] includes a variety of labeling methods that we do not discuss in this survey. The extension of graceful labelings to directed graphs arose in the characterization of finite neofields by Hsu and Keedwell [445], [446]. The relationship between graceful digraphs and a variety of algebraic structures including cyclic difference sets, sequenceable groups, generalized complete mappings, near-complete mappings, and neofields is discussed in [179] and [180]. The connection between graceful labelings and perfect systems of difference sets is given in [162]. Labeled graphs serve as useful models for a broad range of applications such as: coding theory, x-ray crystallography, radar, astronomy, circuit design, communication network addressing, data base management, and models for constraint programming over finite domains–see [176], [177], [895], [710], [855], [856], and [648] for details. Terms and notation not defined below follow that used in [237] and [346].

2 Graceful and Harmonious Labelings

2.1 Trees

The Ringel-Kotzig conjecture that all trees are graceful has been the focus of many papers. Kotzig [448] has called the effort to prove it a “disease.” Among the trees known to be graceful are: caterpillars [733] (a caterpillar is a tree with the property that the removal of its endpoints leaves a path); trees with at most 4 end-vertices [448], [1009] and [462]; trees with diameter at most 5 [1009] and [442]; trees with at most 27 vertices [38]; symmetrical trees (i.e., a rooted tree in which every level contains vertices of the same degree) [163], [708]; rooted trees where the roots have odd degree and the lengths of the paths from the root to the leaves differ by at most one and all the internal vertices have the same parity [219]; the graph obtained by identifying the endpoints any number of paths of a fixed length except for the case that the length has the form 4r + 1, r > 1 and the number of paths is of the form 4m with m > r [753]; regular bamboo trees [753] (a rooted tree consisting of branches of equal length the endpoints of which are identified with end points of stars of equal size); and olive trees [698], [2] (a rooted tree consisting of k branches, where the ith branch is a path of length i). Aldred, Širáň and Širáň [39] have proved that the number of graceful labelings of P_{n} grows at least as fast as (5/3)^{n}. They mention that this fact has an application to topological graph theory. One such application was provided by Goddyn, Richter, and and Širáň [369] who used graceful labelings of paths on 2s + 1 vertices (s ≥ 2) to obtain 2^{2s} cyclic oriented triangular embeddings of the complete graph on 12s + 7 vertices. The Aldred, Širáň and Širáň bound was improved by Adamaszek [32] to (2.37)^{n} with the aid of a computer. Cattell [227] has shown that when finding a graceful labeling of a path one has almost complete freedom to choose a particular label i for any given vertex v. In particular, he shows that the only cases of P_{n} when this cannot be done are when n ≡ 3 (mod 4) or n ≡ 1 (mod 12), v is in the smaller of the two partite sets of vertices, and i = (n − 1)/2.

A spider is a tree that has at most one vertex (called the center) of degree greater than 2. Bahls, Lake, and Wertheim [114] proved that spiders for which the lengths of every path from the center to a leaf differ by at most one are graceful and for spiders for which the lengths of every path from the center to a leaf has the same length and there is an odd number of such paths there is a family of graceful labelings.

In [312] and [313] Eshghi and Azimi [312] discuss a programming model for finding graceful labelings of large graphs. The computational results show that the models can easily solve the graceful labeling problems for large graphs. They used this method to verify that all trees with 30, 35, or 40 vertices are graceful. Stanton and Zarnke [873] and Koh, Rogers, and Tan [498], [499], [497] gave methods for combining graceful trees to yield larger graceful trees. Rogers in [731] and Koh, Tan, and Rogers in [496] provide recursive constructions to create graceful trees. Burzio and Ferrarese [206] have shown that the graph obtained from any graceful tree by subdividing every edge is also graceful. Morgan [667] has used Skolem sequences to construct classes of graceful trees. In 1979 Bermond [159] conjectured that lobsters are graceful (a lobster is a tree with the property that the removal of the endpoints leaves a caterpillar). Morgan [666] has shown that all lobsters with perfect matchings are graceful. Mishra and Panigrahi [662] and [696] found classes of graceful lobsters of diameter at least five. In [783] Sethuraman and Jesintha [783] explores how one can generate graceful lobsters from a graceful caterpillar while in [787] and [788] (see also Jesin) they show how to generate graceful trees from a graceful star. More special cases of Bermond’s conjecture have been done by Ng [681], by Wang, Jin, Lu, and Zhang [944], and by Abhyanker [1]. Morgan and Rees [668] have used Skolem and Hooked-Skolem sequences to generate classes of graceful lobsters. Whether or not lobsters are harmonious seems to have attracted no attention thus far.

Barrientos [138] defines a y-tree as a graph obtained from a path by appending an edge to a vertex of a path adjacent to an end point. He proves that graphs obtained from a y-tree T by replacing every edge e_{i} of T by a copy of K_{2,n_{i}} in such a way that the ends of e_{i} are merged with the two independent vertices of K_{2,n_{i}} after removing the edge e_{i} from T are graceful.

Bermond and Sotteau [163] have shown that a rooted tree in which every level contains vertices of the same degree (symmetrical trees) are graceful. Sethuraman and Jesintha [784], [785] and [786] (see also [459]) proved that rooted trees obtained by identifying one of the end vertices adjacent to either of the penultimate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. They also proved that rooted trees obtained by identifying either of the penultimate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. In [784], [785], and [786] (see also [459]) Sethuraman and Jesintha prove that all rooted trees in which every level contains pendant vertices and the degrees of the internal vertices in the same level are equal are graceful. Kanetkar and Sane [469] show that trees formed by identifying one end vertex of each of six or fewer paths whose lengths determine an arithmetic progression are graceful.

Chen, Lü, and Yeh [241] define a firecracker as a graph obtained from the concatenation of stars by linking one leaf from each. They also define a banana tree as a graph obtained by connecting a vertex v to one leaf of each of any number of stars (v is not in any of the stars). They proved that firecrackers are graceful and conjecture that banana trees are graceful. Sethuraman and Jesintha [790] and [789] (see also [459] have shown that all banana trees and extended banana trees (graphs obtained by joining a vertex to one leaf of each of any number of stars by a path of length of at least two) are graceful. Various kinds of bananas trees had been shown to be graceful by Bhat-Nayak and Deshmukh [165], by Murugan and Arumugam [675], [673] and by Vilfred [925]. Despite the efforts of many, the graceful tree conjecture remains open even for trees with maximum degree 3.

Aldred and McKay [38] used a computer to show that all trees with at most 26 vertices are harmonious. That caterpillars are harmonious has been shown by Graham and Sloane [380]. In a paper published in 2004 Krishnaa [516] claims to proved that all trees have both graceful and harmonious labelings. However, her proofs were flawed.

Using a variant of the Matrix Tree Theorem, Whitty [953] specifies an n × n matrix of indeterminates whose determinant is a multivariate polynomial that enumerates the gracefully labelled (n + 1)-vertex trees. Whitty also gives a bijection between gracefully labelled graphs and rook placements on a chessboard on the Möbius strip.

Cahit extended the notion of gracefulness to directed graphs in [220]. More specialized results about trees are contained in [159], [175], [495], [629], [213], [461], and [734].

2.2 Cycle-Related Graphs

Cycle-related graphs have been the major focus of attention. Rosa [733] showed that the n-cycle C_{n} is graceful if and only if n ≡ 0 or 3 (mod 4) and Graham and Sloane [380] proved that C_{n} is harmonious if and only if n ≡ 1 or 3 (mod 4). Wheels W_{n} = C_{n} + K_{1} are both graceful and harmonious – [334], [440] and [380]. As a consequence we have that a subgraph of a graceful (harmonious) graph need not be graceful (harmonious). The n-cone (also called the n-point suspension of C_{m}) C_{m} + \overline{K_{n}} has been shown to be graceful when m ≡ 0 or 3 (mod 12) by Bhat-Nayak and Selvam [171]. When n is even and m is 2, 6 or 10 (mod 12) C_{m} + \overline{K_{n}} violates the parity condition for a graceful graph. Bhat-Nayak and Selvam [171] also prove that the following cones are graceful: C_{4} + \overline{K_{n}}, C_{5} + \overline{K_{2}}, C_{7} + \overline{K_{n}}, C_{9} + \overline{K_{2}}, C_{11} + \overline{K_{n}} and C_{19} + \overline{K_{n}}. The helm H_{n} is the graph obtained from a wheel by attaching a pendant edge at each vertex of the n-cycle. Helms have been shown to be graceful [55] and harmonious [367], [616], [617] (see also [620], [772], [608], [271] and [715]). Koh, Rogers, Teo, and Yap, [500] define a web graph as one obtained by joining the pendant points of a helm to form a cycle and then adding a single pendant edge to each vertex of this outer cycle. They asked whether such graphs are graceful. This was proved by Kang, Liang, Gao, and Yang [472]. Yang has extended the notion of a web by iterating the process of adding pendant points and joining them to form a cycle and then adding pendant points to the new cycle. In his notation, W (2, n) is the web graph whereas W (t, n) is the generalized web with t n-cycles. Yang has shown that W (3, n) and W (4, n) are graceful (see [472]), Abhyanker and Bhat-Nayak [3] have done W (5, n) and Abhyanker [1] has done W (t, 5) for 5 ≤ t ≤ 13. Gnanajothi [367] has shown that webs with odd cycles are harmonious. Seoud and Youssef [772] define a closed helm as the graph obtained from a helm by joining each pendant vertex to form a cycle and a flower as the graph obtained from a helm by joining each pendant vertex to the central vertex of the helm. They prove that closed helms and flowers are harmonious when the cycles are odd. A gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle. In 1984 Ma and Feng [633] proved all gears are graceful while in a Master’s thesis in 2006 Chen [242] proved all gears are harmonious. Liu [616] has shown that if two or more vertices are inserted between every pair of vertices of the n-cycle of the wheel W_{n}, the resulting graph is graceful. Liu [614] has also proved that the graph obtain from a gear graph by attaching one or more pendant points to each vertex between the cycle vertices is graceful.

Abhyanker [1] has investigated various unicyclic (that is, graphs with exactly one cycle) graphs. He proved that the unicyclic graphs obtained by identifying one vertex of C_{4} with the root of the olive tree with 2n branches and identifying an adjacent vertex on C_{4} with the end point of the path P_{2n−2} are graceful. He showed that if one attaches any number of pendent edges to these unicyclic graphs at the vertex of C_{4} that is adjacent to the root of the olive tree but not adjacent to the end vertex of the attached path, the resulting graphs are graceful. Likewise, Abhyanker proved that the graph obtained by deleting the branch of length 1 from an olive tree with 2n branches and identifying the root of the edge deleted tree with a vertex of a cycle of the form C_{2n+3} is graceful. He also has a number of results similar to these.

Delorme, Maheo, Thuillier, Koh, and Teo [274] and Ma and Feng [632] showed that any cycle with a chord is graceful. This was first conjectured by Bodendiek, Schumacher, and Wegner [182], who proved various special cases. In 1985 Koh and Yap [501] generalized this by defining a cycle with a P_{k}-chord to be a cycle with the path P_{k} joining two nonconsecutive vertices of the cycle. They proved that these graphs are graceful when k = 3 and conjectured that all cycles with a P_{k}-chord are graceful. This was proved for k ≥ 4 by Punnim and Pabhapote in 1987 [711]. Chen [247] obtained the same result except for three cases which were then handled by Gao [395]. In 2005, Sethuraman and Elmalai [778] defined a cycle with parallel P_{k}-chords as a graph obtained from a cycle C_{n} (n ≥ 6) with consecutive vertices v_{0}, v_{1}, …, v_{n−1} by adding a disjoint path P_{k}, (k ≥ 3), between each pair of nonadjacent vertices v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{i}v_{n−i}, …, v_{α}v_{β} where α = ⌊n/2⌋ − 1 and β = ⌊n/2⌋ + 2 if n is odd or β = ⌊n/2⌋ + 1 if n is even. They proved that every cycle C_{n} (n ≥ 6) with parallel P_{k}-chords is graceful for k = 3, 4, 6, 8, and 10 and they conjecture that the cycle C_{n} with parallel P_{k}-chords is graceful for all even k. Xu [971] proved that all cycles with a chord are harmonious except for C_{6} in the case where the distance in C_{6} between the endpoints of the chord is 2. The gracefulness of cycles with consecutive chords have also been investigated. For 3 ≤ p ≤ n − r, let C_{n}(p, r) denote the n-cycle with consecutive vertices v_{1}, v_{2}, …, v_{n} to which the r chords v_{1}v_{p}, v_{1}v_{p+1}, …, v_{1}v_{p+r−1} have been added. Koh and Punnin [491] and Koh, Rogers, Teo, and Yap [500] have handled the cases r = 2, 3 and n−3 where n is the length of the cycle. Goh and Lim [370] then proved that all remaining cases are graceful. Moreover, Ma [631] has shown that C_{n}(p, n − p) is graceful when p ≡ 0, 3 (mod 4) and Ma, Liu, and Liu [634] have proved other special cases of these graphs are graceful. Ma also proved that if one adds to the graph C_{n}(3, n − 3) any number k_{i} of paths of length 2 from the vertex v_{1} to the vertex v_{i} for i = 2, …, n, the resulting graph is graceful. Chen [247] has shown that apart from four exceptional cases, a graph consisting of three independent paths joining two vertices of a cycle is graceful. This generalizes the result that a cycle plus a chord is graceful. Liu [613] has shown that the n-cycle with consecutive vertices v_{1}, v_{2}, …, v_{n} to which the chords v_{1}v_{k} and v_{1}v_{k+2} (2 ≤ k ≤ n − 3) are adjoined is graceful.

In [272] Deb and Limaye use the notation C(n, k) to denote the cycle C_{n} with k cords sharing a common endpoint called the apex. For certain choices of n and k there is a unique C(n, k) graph and for other choices there is more than one graph possible. They call these shell-type graphs and they call the unique graph C(n, n−3) a shell. Notice that the shell C(n, n − 3) is the same as the fan F_{n−1} = P_{n−1} + K_{1}. Deb and Limaye define a multiple shell to be a collection of edge disjoint shells that have their apex in common. A multiple shell is said to be balanced with width w if every shell has order w or every shell has order w or w + 1. Deb and Limaye [272] have conjectured that all multiple shells are harmonious, and have shown that the conjecture is true for the balanced double shells and balanced triple shells. Yang, Xu, Xi, and Qiao [988] proved the conjecture is true for balanced quadruple shells.

Sethuraman and Dhavamani [775] use H(n, t) to denote the graph obtained from the cycle C_{n} by adding t consecutive chords incident with a common vertex. If the common vertex is u and v is adjacent to u, then for k ≥ 1, n ≥ 4 and 1 ≤ t ≤ n − 3, Sethuraman and Dhavamani denote by G(n, t, k) the graph obtained by taking the union of k copies of H(n, k) with the edge uv identified. They conjecture that every graph G(n, t, k) is graceful. They prove the conjecture for the case that t = n − 3.

For i = 1, 2, …, n let v_{i,1}, v_{i,2}, …, v_{i,2m} be the successive vertices of n copies of C_{2m}. Sekar [753] defines a chain of cycles C_{2m,n} as the graph obtained by identifying v_{i,m} and v_{i+1,m} for i = 1, 2, …, n − 1. He proves that C_{6,2k} and C_{8,n} are graceful for all k and all n. Barrientos [141] proved that all C_{8,n}, C_{12,n}, and C_{6,2k} are graceful.

Truszczyński [918] studied unicyclic graphs and proved several classes of such graphs are graceful. Among these are what he calls dragons. A dragon is formed by joining the end point of a path to a cycle (Koh, et al. [500] call these tadpoles; Kim and Park [487] call them kites). This work led Truszczyński to conjecture that all unicyclic graphs except C_{n}, where n ≡ 1 or 2 (mod 4), are graceful. Guo [394] has shown that dragons are graceful when the length of the cycle is congruent to 1 or 2 (mod 4). In his Master’s thesis, Doma [283] investigates the gracefulness of various unicyclic graphs where the cycle has up to 9 vertices. Because of the immense diversity of unicyclic graphs, a proof of Truszczyński’s conjecture seems out of reach in the near future.

Cycles that share a common edge or a vertex have received some attention. Murugan and Arumugan [674] have shown that books with n pentagonal pages (i.e., n copies of C_{5} with an edge in common) are graceful when n is even and not graceful when n is odd. Let C_{n}^{(t)} denote the one-point union of t cycles of length n. Bermond, Brouwer, and Germa [160] and Bermond, Kotzig, and Turgeon [162]) proved that C_{3}^{(t)} (that is, the friendship graph or Dutch t-windmill) is graceful if and only if t ≡ 0 or 1 (mod 4) while Graham and Sloane [380] proved C_{3}^{(t)} is harmonious if and only if t ≢ 2 (mod 4). Koh, Rogers, Lee, and Toh [492] conjecture that C_{n}^{(t)} is graceful if and only if nt ≡ 0 or 3 (mod 4). Yang and Lin [980] have proved the conjecture for the case n = 5 and Yang, Xu, Xi, Li and Haque [986] did the case n = 7. Xu, Yang, Li and Xi [973] did the case n = 11. Qian [713] verifies this conjecture for the case that t = 2 and n is even and Yang, Xu, Xi, and Li [987] did the case n = 9. Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] have shown that if m ≡ 0 (mod 4) then the one-point union of 2, 3 or 4 copies of C_{m} admits a special kind of graceful labeling called an α-labeling (see Section 3.1) and if m ≡ 2 (mod 4), then the one-point union of 2 or 4 copies of C_{m} admits an α-labeling. Bodendiek, Schumacher, and Wegner [188] proved that the one-point union of any two cycles is graceful when the number of edges is congruent to 0 or 3 modulo 4. (The other cases violate the necessary parity condition.) Shee [803] has proved that C_{4}^{(t)} is graceful for all t. Seoud and Youssef [770] have shown that the one-point union of a triangle and C_{n} is harmonious if and only if n ≡ 1 (mod 4) and that if the one-point union of two cycles is harmonious then the number of edges is divisible by 4. The question of whether this latter condition is sufficient is open. Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] have shown that if G is harmonious then the one-point union of an odd number of copies of G using the vertex labeled 0 as the shared point is harmonious. Sethuraman and Selvaraju [797] have shown that for a variety of choices of points, the one-point union of any number of non-isomorphic complete bipartite graphs is graceful. They raise the question of whether this is true for all choices of the common point.

Another class of cycle-related graphs is that of triangular cacti. The block-cutpoint graph of a graph G is a bipartite graph in which one partite set consists of the cut vertices of G, and the other has a vertex b_{i} for each block B_{i} of G. A block of a graph is a maximal connected subgraph that has no cut-vertex. A triangular cactus is a connected graph all of whose blocks are triangles. A triangular snake is a triangular cactus whose block-cutpoint-graph is a path (a triangular snake is obtained from a path v_{1}, v_{2}, …, v_{n} by joining v_{i} and v_{i+1} to a new vertex w_{i} for i = 1, 2, …, n − 1). Rosa [735] conjectured that all triangular cacti with t ≡ 0 or 1 (mod 4) blocks are graceful. (The cases where t ≡ 2 or 3 (mod 4) fail to be graceful because of the parity condition.) Moulton [669] proved the conjecture for all triangular snakes. A proof of the general case (i.e., all triangular cacti) seems hopelessly difficult. Liu and Zhang [620] gave an incorrect proof that triangular snakes with an odd number of triangles are harmonious whereas triangular snakes with n ≡ 2 (mod 4) triangles are not harmonious. Xu [972] subsequently proved that triangular snakes are harmonious if and only if the number of triangles is not congruent to 2 (mod 4).

A double triangular snake consists of two triangular snakes that have a common path. That is, a double triangular snake is obtained from a path v_{1}, v_{2}, …, v_{n} by joining v_{i} and v_{i+1} to a new vertex w_{i} for i = 1, 2, …, n−1 and to a new vertex u_{i} for i = 1, 2, …, n−1. Xi, Yang, and Wang [968] proved that all double triangular snakes are harmonious.

Defining an n-polygonal snake analogous to triangular snakes, Sekar [753] has shown that such graphs are graceful when n ≡ 0 (mod 4) (n ≥ 8) and when n ≡ 2 (mod 4) and the number of polygons is even. Gnanajothi [367, pp. 31–34] had earlier shown that quadrilateral snakes are graceful. Grace [378] has proved that K_{4}-snakes are harmonious. Rosa [735] has also considered analogously defined quadrilateral and pentagonal cacti and examined small cases.

Barrientos [133] calls a graph a kC_{n}-snake if it is a connected graph with k blocks whose block-cutpoint graph is a path and each of the k blocks is isomorphic to C_{n}. (When n > 3 and k > 3 there is more than one kC_{n}-snake.) If a kC_{n}-snake where the path of minimum length that contains all the cut-vertices of the graph has the property that the distance between any two consecutive cut-vertices is ⌊n/2⌋ it is called linear. Barrientos proves that kC_{4}-snakes are graceful and that the linear kC_{6}-snakes are graceful when k is even. He further proves that kC_{8}-snakes and kC_{12}-snakes are graceful in the cases where the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph are all even and that certain cases of kC_{4n}-snakes and kC_{5n}- snakes are graceful (depending on the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph).

Several people have studied cycles with pendant edges attached. Frucht [334] proved that any cycle with a pendant edge attached at each vertex (i.e., a crown) is graceful. Bu, Zhang, and He [204] have shown that any cycle with a fixed number of pendant edges adjoined to each vertex is graceful. Barrientos [137] defines a hairy cycle as a unicyclic graph other than a cycle in which the deletion of any edge of the cycle results in a caterpillar. He proves that all hairy cycles are graceful [137]. This subsumes the result of Bu, Zhang, and He.

If G has order n, the corona of G with H, G ⊙ H is the graph obtained by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H. Barrientos [137] also proves: if G is a graceful graph of order m and size m − 1, then G ⊙ nK_{1} and G + nK_{1} are graceful; if G is a graceful graph of order p and size q with q > p, then (G ∪ (q + 1 − p)K_{1}) ⊙ nK_{1} is graceful; and all unicyclic graphs other than a cycle for which the deletion of any edge from the cycle results in a caterpillar are graceful.

In [135] Barrientos proved that helms (graphs obtained from a wheel by attaching one pendant edge to each vertex) are graceful. Grace [377] showed that an odd cycle with one or more pendant edges at each vertex is harmonious and conjectured that C_{2n} ⊙ K_{1}, an even cycle with one pendant edge attached at each vertex, is harmonious. This conjecture has been proved by Liu and Zhang [619], Liu [616] and [617], Hegde [416], Huang [447], and Bu [195]. Sekar [753] has shown that the graph C_{m} ⊙ P_{n} obtained by attaching the path P_{n} to each vertex of C_{m} e is graceful. For any n ≥ 3 and any t with 1 ≤ t ≤ n, let C_{n}^{+t} denote the class of graphs formed by adding a single pendant edge to t vertices of a cycle of length n. Ropp [732] proved that for every n and t the class C_{n}^{+t} contains a graceful graph. Gallian and Ropp [346] conjectured that for all n and t, all members of C_{n}^{+t} are graceful. This was proved by Qian [713] and by Kang, Liang, Gao, and Yang [472]. Of course, this is just a special case of the aforementioned conjecture of Truszczyński that all unicyclic graphs except C_{n} for n ≡ 1 or 2 (mod 4) are graceful. Sekar [753] proved that the graph obtained by identifying an endpoint of a star with a vertex of a cycle is graceful.

2.3 Product Related Graphs

Graphs that are cartesian products and related graphs have been the subject of many papers. That planar grids, P_{m} × P_{n}, are graceful was proved by Acharya and Gill [20] in 1978 although the much simpler labeling scheme given by Maheo [640] in 1980 for P_{m} ×P_{2} readily extends to all grids. In 1980 Graham and Sloane [380] proved ladders, P_{m} × P_{2}, are harmonious when m > 2 and in 1992 Jungreis and Reid [468] showed that the grids P_{m}×P_{n} are harmonious when (m, n) ≠ (2, 2). A few people have looked at graphs obtained from planar grids in various ways. Kathiresan [475] has shown that graphs obtained from ladders by subdividing each step exactly once are graceful and that graphs obtained by appending an edge to each vertex of a ladder are graceful [477]. Acharya [14] has shown that certain subgraphs of grid graphs are graceful. Lee [531] defines a Mongolian tent as a graph obtained from P_{m} × P_{n}, n odd, by adding one extra vertex above the grid and joining every other vertex of the top row of P_{m} × P_{n} to the new vertex. A Mongolian village is a graph formed by successively amalgamating copies of Mongolian tents with the same number of rows so that adjacent tents share a column. Lee proves that Mongolian tents and villages are graceful. A Young tableau is a subgraph of P_{m} × P_{n} obtained by retaining the first two rows of P_{m} × P_{n} and deleting vertices from the right hand end of other rows in such a way that the lengths of the successive rows form a nonincreasing sequence. Lee and Ng [547] have proved that all Young tableaus are graceful. Lee [531] has also defined a variation of Mongolian tents by adding an extra vertex above the top row of a Young tableau and joining every other vertex of that row to the extra vertex. He proves these graphs are graceful.

Prisms are graphs of the form C_{m} × P_{n}. These can be viewed as grids on cylinders. In 1977 Bodendiek, Schumacher, and Wegner [182] proved that C_{m} × P_{2} is graceful when m ≡ 0 (mod 4). According to the survey by Bermond [159], Gangopadhyay and Rao Hebbare did the case that m is even about the same time. In a 1979 paper, Frucht [334] stated without proof that he had done all C_{m}×P_{2}. A complete proof of all cases and some related results were given by Frucht and Gallian [337] in 1988. In 1992 Jungreis and Reid [468] proved that all C_{m} × P_{n} are graceful when m and n are even or when m ≡ 0 (mod 4). Yang and Wang have shown that the prisms C_{4n+2} × P_{4m+3} [985], C_{n} × P_{2} [983], and C_{6} × P_{m}(m ≥ 2) (see [985]) are graceful. Singh [833] proved that C_{3} × P_{n} is graceful for all n. In their 1980 paper Graham and Sloane [380] proved that C_{m} × P_{n} is harmonious when n is odd and they used a computer to show C_{4} ×P_{2}, the cube, is not harmonious. In 1992 Gallian, Prout, and Winters [350] proved that C_{m} × P_{2} is harmonious when m ≠ 4. In 1992, Jungreis and Reid [468] showed that C_{4} × P_{n} is harmonious when n ≥ 3. Huang and Skiena [449] have shown that C_{m} × P_{n} is graceful for all n when m is even and for all n with 3 ≤ n ≤ 12 when m is odd. Abhyanker [1] proved that the graphs obtained from C_{2m+1} × P_{5} by adding a pendent edge to each vextex of an outercycle is graceful.

Torus grids are graphs of the form C_{m} × C_{n} (m > 2, n > 2). Very little success has been achieved with these graphs. The graceful parity condition is violated for C_{m} × C_{n} when m and n are odd and the harmonious parity condition [380, Theorem 11] is violated for C_{m} × C_{n} when m ≡ 1, 2, 3 (mod 4) and n is odd. In 1992 Jungreis and Reid [468] showed that C_{m} × C_{n} is graceful when m ≡ 0 (mod 4) and n is even. A complete solution to both the graceful and harmonious torus grid problems will most likely involve a large number of cases.

There has been some work done on prism-related graphs. Gallian, Prout, and Winters [350] proved that all prisms C_{m} ×P_{2} with a single vertex deleted or single edge deleted are graceful and harmonious. The Möbius ladder M_{n} is the graph obtained from the ladder P_{n} × P_{2} by joining the opposite end points of the two copies of P_{n}. In 1989 Gallian [345] showed that all Möbius ladders are graceful and all but M_{3} are harmonious. Ropp [732] has examined two classes of prisms with pendant edges attached. He proved that all C_{m} × P_{2} with a single pendant edge at each vertex are graceful and all C_{m} × P_{2} with a single pendant edge at each vertex of one of the cycles are graceful.

Another class of cartesian products that has been studied is that of books and “stacked” books. The book B_{m} is the graph S_{m} × P_{2} where S_{m} is the star with m + 1 vertices. In 1980 Maheo [640] proved that the books of the form B_{2m} are graceful and conjectured that the books B_{4m+1} were also graceful. (The books B_{4m+3} do not satisfy the graceful parity condition.) This conjecture was verified by Delorme [273] in 1980. Maheo [640] also proved that L_{n} ×P_{2} and B_{2m} ×P_{2} are graceful. Both Grace [376] and Reid (see [349]) have given harmonious labelings for B_{2m}. The books B_{4m+3} do not satisfy the harmonious parity condition [380, Theorem 11]. Gallian and Jungreis [349] conjectured that the books B_{4m+1} are harmonious. Gnanajothi [367] has verified this conjecture by showing B_{4m+1} has an even stronger form of labeling – see Section 4.1. Liang [596] also proved the conjecture. In 1988 Gallian and Jungreis [349] defined a stacked book as a graph of the form S_{m} × P_{n}. They proved that the stacked books of the form S_{2m} × P_{n} are graceful and posed the case S_{2m+1} × P_{n} as an open question. The n-cube K_{2} × K_{2} × ⋯ × K_{2} (n copies) was shown to be graceful by Kotzig [507]—see also [640]. In 1986 Reid [722] found a harmonious labeling for K_{4} × P_{n}. Petrie and Smith [700] have investigated graceful labelings of graphs as an exercise in constraint programming satisfaction. They have shown that K_{m} × P_{n} is graceful for (m, n) = (4, 2), (4, 3), (4, 4), (4, 5), and (5,2) but is not graceful for (3, 3) and (6, 2). Their labeling for K_{5} × P_{2} is the unique graceful labeling. They also considered the graph obtained by identifying the hubs of two copies of W_{n}. The resulting graph is not graceful when n = 3 but is graceful when n is 4 and 5. Smith [855] has used a computer search to prove that K_{m} × P_{2} is not graceful for m = 7, 8, 9, and 10. She conjectures that K_{m} × P_{2} is not graceful for m > 5.

For a bipartite graph G with partite sets X and Y let G′ be a copy of G and X′ and Y′ be copies of X and Y. Lee and Liu [543] define the mirror graph, M(G), of G as the disjoint union of G and G′ with additional edges joining each vertex of Y to its corresponding vertex in Y′. The case that G = K_{m,n} is more simply denoted by M(m, n). They proved that for many cases M(m, n) has a stronger form of graceful labeling (see §3.1 for details).

The composition G_{1}[G_{2}] is the graph having vertex set V (G_{1}) × V (G_{2}) and edge set {(x_{1}, y_{1}), (x_{2}, y_{2})| x_{1}x_{2} ∈ E(G_{1}) or x_{1} = x_{2} and y_{1}y_{2} ∈ E(G_{2})}. The symmetric product G_{1} ⊕ G_{2} of graphs G_{1} and G_{2} is the graph with vertex set V (G_{1}) × V (G_{2}) and edge set {(x_{1}, y_{1}), (x_{2}, y_{2})| x_{1}x_{2} ∈ E(G_{1}) or y_{1}y_{2} ∈ E(G_{2}) but not both}. Seoud and Youssef [771] have proved that P_{n} ⊕ \overline{K_{2}} is graceful when n > 1 and P_{n}[P_{2}] is harmonious for all n. They also observe that the graphs C_{m} ⊕ C_{n} and C_{m}[C_{n}] violate the parity conditions for graceful and harmonious graphs when m and n are odd.

2.4 Complete Graphs

The questions of the gracefulness and harmoniousness of the complete graphs K_{n} have been answered. In each case the answer is positive if and only if n ≤ 4 ([371], [832], [380], [164]). Both Rosa [733] and Golomb [371] proved that the complete bipartite graphs K_{m,n} are graceful while Graham and Sloane [380] showed they are harmonious if and only if m or n = 1. Aravamudhan and Murugan [51] have shown that the complete tripartite graph K_{1,m,n} is both graceful and harmonious while Gnanajothi [367, pp. 25–31] has shown that K_{1,1,m,n} is both graceful and harmonious and K_{2,m,n} is graceful. Some of the same results have been obtained by Seoud and Youssef [766] who also observed that when m, n, and p are congruent to 2 (mod 4), K_{m,n,p} violates the parity conditions for harmonious graphs.

Beutner and Harborth [164] show that K_{n} − e (K_{n} with an edge deleted) is graceful only if n ≤ 5, any K_{n} − 2e (K_{n} with two edges deleted) is graceful only if n ≤ 6, and any K_{n} − 3e is graceful only if n ≤ 6. They also determine all graceful graphs K_{n} − G where G is K_{1,a} with a ≤ n − 2 and where G is a matching M_{a} with 2a ≤ n. They give graceful labelings for K_{1,m,n}, K_{2,m,n}, K_{1,1,m,n} and conjecture that these and K_{m,n} are the only complete multipartite graphs that are graceful. They have verified this conjecture for graphs with up to 23 vertices via computer.

The windmill graph K_{n}^{(m)} (n > 3) consists of m copies of K_{n} with a vertex in common. A necessary condition for K_{n}^{(m)} to be graceful is that n ≤ 5 – see [500]. Bermond [159] has conjectured that K_{4}^{(m)} is graceful for all m ≥ 4. This is known to be true for m ≤ 22 [449]. Bermond, Kotzig, and Turgeon [162] proved that K_{n}^{(m)} is not graceful when n = 4 and m = 2 or 3, and when m = 2 and n = 5. In 1982 Hsu [444] proved that K_{4}^{(m)} is harmonious for all m. Graham and Sloane [380] conjectured that K_{n}^{(2)} is harmonious if and only if n = 4. They verified this conjecture for the cases that n is odd or n = 6. Liu [608] has shown that K_{n}^{(2)} is not harmonious if n = 2^{a}p_{1}^{a_{1}} ⋯ p_{s}^{a_{s}} where a, a_{1}, …, a_{s} are positive integers and p_{1}, …, p_{s} are distinct odd primes and there is a j for which p_{j} ≡ 3 (mod 4) and a_{j} is odd. He also shows that K_{n}^{(3)} is not harmonious when n ≡ 0 (mod 4) and 3n = 4^{e}(8k + 7) or n ≡ 5 (mod 8). Koh, Rogers, Lee and Toh [492] and Rajasingh and Pushpam [716] have shown that K_{m,n}^{(t)}, the one-point union of t copies of K_{m,n}, is graceful. Sethuraman and Selvaraju [793] have proved that the one-point union of graphs of the form K_{2,m_{i}} for i = 1, 2, …, n, where the union is taken at a vertex from the partite set with 2 vertices is graceful if at most two of the m_{i} are equal. They conjecture that the restriction that at most two of the m_{i} are equal is not necessary. Koh, Rogers, Lee, and Toh [500] introduced the notation B(n, r, m) for the graph consisting of m copies of K_{n} with a K_{r} in common (n ≥ r). (We note that Guo [395] has used the notation B(n, r, m) to denote three independent paths of lengths n, r and m joining two vertices.) Bermond [159] raised the question: “For which m, n, and r is B(n, r, m) graceful?” Of course, the case r = 1 is the same as K_{n}^{(m)}. For r > 1, B(n, r, m) is graceful in the following cases: n = 3, r = 2, m ≥ 1 [493]; n = 4, r = 2, m ≥ 1 [273]; n = 4, r = 3, m ≥ 1 (see [159]), [493]. Seoud and Youssef [766] have proved B(3, 2, m) and B(4, 3, m) are harmonious. Liu [607] has shown that if there is a prime p such that p ≡ 3 (mod 4) and p divides both n and n − 2 and the highest power of p that divides n and n − 2 is odd, then B(n, 2, 2) is not graceful. Smith [855] has shown that up to symmetry, B(5, 2, 2) has a unique graceful labeling; B(n, 3, 2) is not graceful for n = 6, 7, 8, 9, and 10; B(6, 3, 3) and B(7, 3, 3) are not graceful, and B(5, 3, 3) is graceful. Combining results of Bermond and Farhi [161] and Smith [855] show that B(n, 2, 2) is not graceful for n > 5. More generally, Bermond and Farhi [161] have investigated the class of graphs consisting of m copies of K_{n} having exactly k copies of K_{r} in common. They proved such graphs are not graceful for n sufficiently large compared to r. Barrientos [138] proved that the graph obtained by performing the one-point union of any collection of complete bipartite graphs K_{m_{1},n_{1}}, K_{m_{2},n_{2}}, …, K_{m_{t},n_{t}}, where each K_{m_{i},n_{i}} appears at most twice and gcd(n_{1}, n_{2}, …, n_{t}) = 1, is graceful.

Sethuraman and Elumalai [777] have shown that K_{1,m,n} with a pendent edge attached to each vertex is graceful and Jirimutu [464] has shown that the graph obtained by attaching a pendant edge to every vertex of K_{m,n} is graceful (see also [43]). In [791] Sethuraman and Kishore determine the graceful graphs that are the union of n copies of K_{4} with i edges deleted for 1 ≤ i ≤ 5 with one edge in common. The only cases that are not graceful are those graphs where the members of the union are C_{4} for n ≡ 3 (mod 4) and where the members of the union are P_{2}. They conjecture that these two cases are the only instances of edge induced subgraphs of the union of n copies of K_{4} with one edge in common that are not graceful. Sethuraman and Selvaraju [799] have shown that union of any number of copies of K_{4} with an edge deleted and one edge in common is harmonious. Bhat-Nayah and Gohkale [170] have shown that \overline{K_{n}}+2K_{2} is not graceful whereas Amutha and Kathiresan [43] proved that the graph obtained by a attaching pendent edge to each vertex of \overline{K_{n}} + 2K_{2} is graceful.

Clemens, Coulibaly, Garvens, Gonnering, Lucas, and Winters [263] investigated the gracefulness of the one-point and two-point unions of graphs. They show the following graphs are graceful: the one-point union of an end vertex of P_{n} and K_{4}; the graph obtained by taking the one-point union of K_{4} with one end vertex of P_{n} and the one-point union of the other end vertex of P_{n} with the central vertex of K_{1,r}; the graph obtained by taking the one-point union of K_{4} with one end vertex of P_{n} and the one-point union of the other end of P_{n} with a vertex from the partite set of order 2 of K_{2,r}; the graph obtained from the graph just described by appending any number of edges to the other vertex of the partite set of order 2; the two-point union of the two vertices of the partite set of order 2 in K_{2,r} and two vertices from K_{4}; and the graph obtained from the graph just described by appending any number of edges to one of the vertices from the partite set of order 2.

2.5 Disconnected Graphs

There have been many papers dealing with graphs that are not connected. For any graph G the graph mG denotes the disjoint union of m copies of G. In 1975 Kotzig [506] investigated the gracefulness of the graphs rC_{s}. When rs ≡ 1 or 2 (mod 4), these graphs violate the gracefulness parity condition. Kotzig proved that when r = 3 and 4k > 4, then rC_{4k} has a stronger form of graceful labeling called α-labeling (see §3.1) whereas when r ≥ 2 and s = 3 or 5, rC_{s} is not graceful. In 1984 Kotzig [508] once again investigated the gracefulness of rC_{s} as well as graphs that are the disjoint union of odd cycles. For graphs of the latter kind he gives several necessary conditions. His paper concludes with an elaborate table that summarizes what was then known about the gracefulness of rC_{s}. He [407] has shown that graphs of the form 2C_{2m} and graphs obtained by connecting two copies of C_{2m} with an edge are graceful. Cahit [216] has shown that rC_{s} is harmonious when r and s are odd and Seoud, Abdel Maqsoud, and Sheehan [757] noted that when r or s is even, rC_{s} is not harmonious. Seoud, Abdel Maqsoud, and Sheehan [757] proved that C_{n} ∪ C_{n+1} is harmonious if and only if n ≥ 4. They conjecture that C_{3} ∪ C_{2n} is harmonious when n ≥ 3. This conjecture was proved when Yang, Lu, and Zeng [981] showed that all graphs of the form C_{2j+1} ∪ C_{2n} are harmonious except for (n, j) = (2, 1).

In 1978 Kotzig and Turgeon [511] proved that mK_{n} (i.e., the union of m disjoint copies of K_{n}) is graceful if and only if m = 1 and n ≤ 4. Liu and Zhang [620] have shown that mK_{n} is not harmonious for n odd and m ≡ 2 (mod 4) and is harmonious for n = 3 and m odd. They conjecture that mK_{3} is not harmonious when m ≡ 0 (mod 4). Bu and Cao [196] give some sufficient conditions for the gracefulness of graphs of the form K_{m,n} ∪ G and they prove that K_{m,n} ∪ P_{t} and the disjoint union of complete bipartite graphs are graceful under some conditions.

A Skolem sequence of order n is a sequence s_{1}, s_{2}, …, s_{2n} of 2n terms such that, for each k ∈ {1, 2, …, n}, there exist exactly two subscripts i(k) and j(k) with s_{i(k)} = s_{j(k)} = k and |i(k) − j(k)| = k. A Skolem sequence of order n exists if and only if n ≡ 0 or 1 (mod 4). Abrham [5] has proved that any graceful 2-regular graph of order n ≡ 0 (mod 4) in which all the component cycles are even or of order n ≡ 3 (mod 4), with exactly one component an odd cycle, can be used to construct a Skolem sequence of order n + 1. Also, he showed that certain special Skolem sequences of order n can be used to generate graceful labelings on certain 2-regular graphs.

In 1985 Frucht and Salinas [338] conjectured that C_{s} ∪ P_{n} is graceful if and only if s + n ≥ 7 and they proved the conjecture for the case that s = 4. Frucht [336] did the case the s = 3 and the case that s = 2n + 1. Bhat-Nayak and Deshmukh [168] also did the case s = 3 and they have done the cases of the form C_{2x+1} ∪ P_{x−2θ} where 1 ≤ θ ≤ ⌊(x − 2)/2⌋ [169]. Choudum and Kishore [256] have done the cases where s ≥ 5 and n ≥ (s + 5)/2 and Kishore [490] did the case s = 5. Gao and Liang [353] have done the following cases: s > 4, n = 2 (see also [352]); s = 4k, n = k + 2, n = k + 3, n = 2k + 2; s = 4k + 1, n = 2k, n = 3k − 1, n = 4k − 1; s = 4k + 2, n = 3k, n = 3k + 1, n = 4k + 1; s = 4k + 3, n = 2k + 1, n = 3k, n = 4k. Seoud, Abdel Maqsoud, and Sheehan [760] did the case that s = 2k (k ≥ 3) and n ≥ k + 1 as well as the cases where s = 6, 8, 10, 12 and n ≥ 2. Shimazu [808] has handled the cases that s ≥ 5 and n = 2, s ≥ 4 and n = 3 and s = 2n + 2 and n ≥ 2. Liang [597] has done the following cases: s = 4k, n = k + 2, k + 3, 2k + 1, 2k + 2, 2k + 3, 2k + 4, 2k + 5; s = 4k − 1, n = 2k, 3k − 1, 4k − 1; s = 4k + 2, n = 3k, 3k + 1, 4k + 1; s = 4k + 3, n = 2k + 1, 3k, 4k. Youssef [994] proved that C_{5} ∪ S_{n} is graceful if and only if n = 1 or 2 and that C_{6} ∪ S_{n} is graceful if and only if n is odd or n = 2 or 4.

Seoud and Youssef [773] have shown that K_{5}∪K_{m,n}, K_{m,n}∪K_{p,q} (m, n, p, q ≥ 2), K_{m,n}∪ K_{p,q} ∪ K_{r,s} (m, n, p, q, r, s ≥ 2, (p, q) ≠ (2, 2)), and pK_{m,n} (m, n ≥ 2, (m, n) ≠ (2, 2)) are graceful. They also prove that C_{4} ∪ K_{1,n} (n ≠ 2) is not graceful whereas Choudum and Kishore [258], [490] have proved that C_{s} ∪ K_{1,n} is graceful for every s ≥ 7 and n ≥ 1. Lee, Quach, and Wang [558] established the gracefulness of P_{s} ∪ K_{1,n}. Seoud and Wilson [765] have shown that C_{3} ∪ K_{4}, C_{3} ∪ C_{3} ∪ K_{4}, and certain graphs of the form C_{3} ∪ P_{n} and C_{3} ∪ C_{3} ∪ P_{n} are not graceful. Abrham and Kotzig [10] proved that C_{p} ∪ C_{q} is graceful if and only if p + q ≡ 0 or 3 (mod 4). Zhou [1011] proved that K_{m} ∪ K_{n} (n > 1, m > 1) is graceful if and only if {m, n} = {4, 2} or {5, 2}. (C. Barrientos has called to my attention that K_{1} ∪ K_{n} is graceful if and only if n = 3 or 4.) Shee [802] has shown that graphs of the form P_{2} ∪ C_{2k+1} (k > 1), P_{3} ∪ C_{2k+1}, P_{n} ∪ C_{3}, and S_{n} ∪ C_{2k+1} all satisfy a condition that is a bit weaker than harmonious. Bhat-Nayak and Deshmukh [166] have shown that C_{4t} ∪ K_{1,4t−1} and C_{4t+3} ∪ K_{1,4t+2} are graceful. Section 3.1 includes numerous families of disconnected graphs that have a stronger form of graceful labelings.

In considering graceful labelings of the disjoint unions of two or three stars with e edges Yang and Wang [984] permitted the vertex labels to range from 0 to e + 1 and 0 to e+2, respectively. With these definitions of graceful, they proved that S_{m} ∪S_{n} is graceful if and only if m or n is even and that S_{m} ∪ S_{n} ∪ S_{k} is graceful if and only if at least one of m, n, or k is even (m > 1, n > 1, k > 1).

Seoud and Youssef [769] investigated the gracefulness of specific families of the form G ∪ K_{m,n}. They obtained the following results: C_{3} ∪ K_{m,n} is graceful if and only if m ≥ 2 and n ≥ 2; C_{4} ∪ K_{m,n} is graceful if and only if (m, n) ≠ (1, 1); C_{7} ∪ K_{m,n} and C_{8} ∪ K_{m,n} are graceful for all m and n; mK_{3} ∪ nK_{1,r} is not graceful for all m, n and r; K_{i} ∪ K_{m,n} is graceful for i ≤ 4 and m ≥ 2, n ≥ 2 except for i = 2 and (m, n) = (2, 2); K_{5} ∪ K_{1,n} is graceful for all n; K_{6} ∪ K_{1,n} is graceful if and only if n is different than 1 and 3. Youssef [997] completed the characterization of the graceful graphs of the form C_{n} ∪ K_{p,q} where n ≡ 0 or 3 (mod 4) by showing that for n > 8 and n ≡ 0 or 3 (mod 4), C_{n} ∪ K_{p,q} is graceful for all p and q (see also [136]). Note that when n ≡ 1 or 2 (mod 4) certain cases of C_{n} ∪ K_{p,q} violate the parity condition for gracefulness.

For i = 1, 2, …, m let v_{i,1}, v_{i,2}, v_{i,3}, v_{i,4} be a 4-cycle. Yang and Pan [979] define F_{k,4} to be the graph obtained by identifying v_{i,3} and v_{i+1,1} for i = 1, 2, …, k − 1. They prove that F_{m_{1},4} ∪ F_{m_{2},4} ∪ ⋯ ∪ F_{m_{n},4} is graceful for all n. Pan and Lu [695] have shown that (P_{2} + \overline{K_{n}}) ∪ K_{1,m} and (P_{2} + \overline{K_{n}}) ∪ T_{n} are graceful.

Barrientos [136] has shown the following graphs are graceful: C_{6} ∪K_{1,2n+1}; ⋃_{i=1}^{t} K_{m_{i},n_{i}} for 2 ≤ m_{i} < n_{i}; and C_{m} ∪ ⋃_{i=1}^{t} K_{m_{i},n_{i}} for 2 ≤ m_{i} < n_{i}, m ≡ 0 or 3 (mod 4), m ≥ 11.

Youssef [995] has shown that if G is harmonious then mG and G^{m} are harmonious for all odd m. He asks the question of whether G is harmonious implies G^{m} is harmonious when m ≡ 0 (mod 4).

2.6 Joins of Graphs

A number of classes of graphs that are the join of graphs have been shown to be graceful or harmonious. Acharya [11] proved that if G is a connected graph, then G + \overline{K_{n}} is graceful. Redl [718] showed that the double cone C_{n} + \overline{K_{2}} is graceful for n = 3, 4, 5, 7, 8, 9, 11, and 12 but not graceful for n ≡ 2 (mod 4). Reid [722] proved that P_{n} + \overline{K_{t}} is harmonious. Sethuraman and Selvaraju [798] have shown that P_{n} + K_{2} is harmonious. They ask whether S_{n} +P_{n} or P_{m} +P_{n} is harmonious. Of course, wheels are of the form C_{n} +K_{1} and are graceful and harmonious. In 2006 Chen [242] proved that multiple wheels nC_{m} + K_{1} are harmonious for all n ≢ 0 mod 4. She believes that the n ≢ 0 (mod 4) case is also harmonious. Chen also proved that if H has at least one edge, H + K_{1} is harmonious, and n is odd, then nH + K is harmonious.

Shee [802] has proved K_{m,n} + K_{1} is harmonious and observed that various cases of K_{m,n} + K_{t} violate the harmonious parity condition in [380]. Liu and Zhang [620] have proved that K_{2}+K_{2}+⋯+K_{2} is harmonious. Yuan and Zhu [1004] proved that K_{m,n}+K_{2} is graceful and harmonious. Gnanajothi [367, pp. 80–127] obtained the following: C_{n}+\overline{K_{2}} is harmonious when n is odd and not harmonious when n ≡ 2, 4, 6 (mod 8); S_{n} + \overline{K_{t}} is harmonious; and P_{n} + \overline{K_{t}} is harmonious. Balakrishnan and Kumar [122] have proved that the join of \overline{K}_{n} and two disjoint copies of K_{2} is harmonious if and only if n is even. Ramírez-Alfonsín [717] has proved that if G is graceful and |V (G)| = |E(G)| = e and either 1 or e is not a vertex label then G + \overline{K_{t}} is graceful for all t.

Seoud and Youssef [771] have proved: the join of any two stars is graceful and harmonious; the join of any path and any star is graceful; and C_{n} + \overline{K_{t}} is harmonious for every t when n is odd. They also prove that if any edge is added to K_{m,n} the resulting graph is harmonious if m or n is at least 2. Deng [275] has shown certain cases of C_{n} + \overline{K_{t}} are harmonious. Seoud and Youssef [768] proved: the graph obtained by appending any number of edges from the two vertices of degree n ≥ 2 in K_{2,n} is not harmonious; dragons D_{m,n} (i.e., P_{m} is appended to C_{n}) are not harmonious when m+n is odd; and the disjoint union of any dragon and any number of cycles is not harmonious when the resulting graph has odd order. Youssef [994] has shown that if G is a graceful graph with p vertices and q edges with p = q + 1, then G + S_{n} is graceful.

Sethuraman and Elumalai [781] have proved that for every graph G with p vertices and q edges the graph G+K_{1}+\overline{K_{m}} is graceful when m ≥ 2^{p}−p−1−q. As a corollary they deduce that every graph is a vertex induced subgraph of a graceful graph. Balakrishnan and Sampathkumar [123] ask for which m ≥ 3 is the graph \overline{K_{n}} + mK_{2} graceful for all n. Bhat-Nayak and Gokhale [170] have proved that \overline{K_{n}} + 2K_{2} is not graceful. Youssef [994] has shown that \overline{K_{n}} + mK_{2} is graceful if m ≡ 0 or 1 (mod 4) and that \overline{K_{n}} + mK_{2} is not graceful if n is odd and m ≡ 2 or 3 (mod 4).

Wu [962] proves that if G is a graceful graph with n edges and n + 1 vertices then the join of G and \overline{K_{m}} and the join of G and any star are graceful.

2.7 Miscellaneous Results

It is easy to see that P_{n}^{2} is harmonious [377] while a proof that P_{n}^{2} is graceful has been given by Kang, Liang, Gao, and Yang [472]. (P_{n}^{k}, the kth power of P_{n}, is the graph obtained from P_{n} by adding edges that join all vertices u and v with d(u, v) = k.) This latter result proved a conjecture of Grace [377]. Seoud, Abdel Maqsoud, and Sheeham [757] proved that P_{n}^{3} is harmonious and conjecture that P_{n}^{k} is not harmonious when k > 3. The same conjecture was made by Fu and Wu [340]. However, Youssef [1000] has proved that P_{8}^{4} is harmonious and P_{n}^{k} is harmonious when k is odd. Selvaraju [754] has shown that P_{n}^{3} and the graphs obtained by joining the centers of any two stars with the end vertices of the path of length n in P_{n}^{3} are harmonious. Gnanajothi [367, p. 50] has shown that the graph that consists of n copies of C_{6} that have exactly P_{4} in common is graceful if and only if n is even. For a fixed n, let v_{i1}, v_{i2}, v_{i3} and v_{i4} (1 ≤ i ≤ n) be consecutive vertices of n 4-cycles. Gnanajothi [367, p. 35] also proves that the graph obtained by joining each v_{i1} to v_{i+1,3} is graceful for all n and the generalized Petersen graph P (n, k) is harmonious in all cases (see also [563]). Recall P (n, k), where n ≥ 5 and 1 ≤ k ≤ n, has vertex set {a_{0}, a_{1}, …, a_{n−1}, b_{0}, b_{1}, …, b_{n−1}} and edge set {a_{i}a_{i+1} | i = 0, 1, …, n − 1} ∪ {a_{i}b_{i} | i = 0, 1, …, n − 1} ∪ {b_{i}b_{i+k} | i = 0, 1, …, n − 1} where all subscripts are taken modulo n [949]. The standard Petersen graph is P (5, 2).) Redl [718] has shown that P (n, k) is graceful for n = 5, 6, 7, 8, 9, and 10. Vietri [924] proved that P (8t, 3) is graceful for all t. He conjectures that P (8t, 3) have a stronger form a graceful labeling called an α-labeling (see §3.1). The gracefulness of the generalized Petersen graphs appears to be an open problem. A conjecture in the graph theory book by Chartrand and Lesniak [237, p. 266] that graceful graphs with arbitrarily large chromatic numbers do not exist was shown to be false by Acharya, Rao, and Arumugam [26].

Yuan and Zhu [1004] proved that P_{n}^{2k} is harmonious when 1 ≤ k ≤ (n − 1)/2 and that P_{n}^{2k} has a stronger form of harmonious labeling (see Section 4.1) when 2k−1 ≤ n ≤ 4k−1. Cahit [216] proves that the graphs obtained by joining p disjoint paths of a fixed length k to single vertex are harmonious when p is odd and when k = 2 and p is even.

Sethuraman and Selvaraju [792] define a graph H to be a supersubdivision of a graph G, if every edge uv of G is replaced by K_{2,m} (m may vary for each edge) by identifying u and v with the two vertices in K_{2,m} that form one of the two partite sets. Sethuraman and Selvaraju prove that every supersubdivision of a path is graceful and every cycle has some supersubdivision that is graceful. They conjecture that every supersubdivision of a star is graceful and that paths and stars are the only graphs for which every supersubdivision is graceful. Barrientos [138] disproved this conjecture by proving that every supersubdivision of a y-trees is graceful (recall a y-tree is obtained from a path by appending an edge to a vertex of a path adjacent to an end point). Barrientos asks if paths and y-trees are the only graphs for which every supersubdivsion is graceful. This seems unlikely to be the case. The conjecture that every supersubdivision of a star is graceful was proved by Kathiresan and Amutha [479]. In [796] Sethuraman and Selvaraju prove that every connected graph has some supersubdivision that is graceful. They pose the question as to whether this result is valid for disconnected graphs. They also ask if there is any graph other than K_{2,m} that can be used to replace an edge of a connected graph to obtain a supersubdivision that is graceful. In [795] Sethuraman and Selvaraju present an algorithm that permits one to start with any non-trivial connected graph and successively form supersubdivisions that have a strong form of graceful labeling called an α-labeling (see §3.1 for teh definition).

Kathiresan [476] uses the notation P_{a,b} to denote the graph obtained by identifying the end points of b internally disjoint paths each of length a. He conjectures that P_{a,b} is graceful except when a is odd and b ≡ 2 (mod 4). He proves the conjecture for the case that a is even and b is odd. Sekar [753] has shown that P_{a,b} is graceful when a ≠ 4r + 1, r > 1; b = 4m, m > r. Yang proved that P_{a,b} is graceful when a = 3, 5, 7, and 9 and b is even and when a = 2, 4, 6, and 8 and b is even (see [982]). Yang, Rong, and Xu [982] proved that P_{a,b} is graceful when a = 10, 12, and 14 and b is even. Kathiresan also shows that the graph obtained by identifying a vertex of K_{n} with any noncenter vertex of the star with 2^{n−1} − n(n − 1)/2 edges is graceful.

For a family of graphs G_{1}(u_{1}, u_{2}), G_{2}(u_{2}, u_{3}), …, G_{m}(u_{m}, u_{m+1}) Cheng, Yao, Chen, and Zhang [250] define a graph-block chain H_{m} as the graph obtained by identifying u_{i+1} of G_{i} with u_{i+1} of G_{i+1} for i = 1, 2, …, m. They denote this graph by H_{m} = G_{1}(u_{1}, u_{2}) ⊕ G_{2}(u_{2}, u_{3}) ⊕ ⋯ ⊕ G_{m}(u_{m}, u_{m+1}). The case where each G_{i} has the form P_{a_{i},b_{i}} they call a path-block chain. The vertex u_{1} is called the initial vertex of H_{m}. They define a generalized spider S_{m}^{∗} as a graph obtained by starting with an initial vertex u_{0} and m path-block graphs and join u_{0} with each initial vertex of each of the path-block graphs. Similarly, they define a generalized caterpillar T_{m}^{∗} as a graph obtained by starting with m path-block chains H_{1}, H_{2}, …, H_{m} and a caterpillar T with m isolated vertices v_{1}, v_{2}, …, v_{m} and join each v_{i} with the initial vertex of each H_{i}. They prove several classes of path-block chains, generalized spiders, and generalized caterpillars are graceful.

The graph T_{n} with 3n vertices and 6n − 3 edges is defined as follows. Start with a triangle T_{1} with vertices v_{1,1}, v_{1,2} and v_{1,3}. Then T_{i+1} consists of T_{i} together with three new vertices v_{i+1,1}, v_{i+1,2}, v_{i+1,3} and edges v_{i+1,1}v_{i,2}, v_{i+1,1}v_{i,3}, v_{i+1,2}v_{i,1}, v_{i+1,2}v_{i,3}, v_{i+1,3}v_{i,1}, v_{i+1,3}v_{i,2}. Gnanajothi [367] proved that T_{n} is graceful if and only if n is odd. Sekar [753] proved T_{n} is graceful when n is odd and T_{n} with a pendant edge attached to the starting triangle is graceful when n is even.

For a graph G, the splitting graph of G, S^{1}(G), is obtained from G by adding for each vertex v of G a new vertex v^{1} so that v^{1} is adjacent to every vertex that is adjacent to v. Sekar [753] has shown that S^{1}(P_{n}) is graceful for all n and S^{1}(C_{n}) is graceful for n ≡ 0, 1 mod 4.

The total graph T (P_{n}) has vertex set V (P_{n}) ∪ E(P_{n}) with two vertices adjacent whenever they are neighbors in P_{n}. Balakrishnan, Selvam, and Yegnanarayanan [124] have proved that T (P_{n}) is harmonious.

For any graph G with vertices v_{1}, …, v_{n} and a vector m = (m_{1}, …, m_{n}) of positive integers the corresponding replicated graph, R_{m}(G), of G is defined as follows. For each v_{i} form a stable set S_{i} consisting of m_{i} new vertices i = 1, 2, …, n (recall a stable set S consists of a set of vertices such that there is not an edge v_{i}v_{j} for all pairs v_{i}, v_{j} in S); two stable sets S_{i}, S_{j}, i ≠ j, form a complete bipartite graph if each v_{i}v_{j} is an edge in G and otherwise there are no edges between S_{i} and S_{j}. Ramírez-Alfonsín [717] has proved that R_{m}(P_{n}) is graceful for all m and all n > 1 (see §3.2 for a stronger result) and that R_{(m,1,...,1)}(C_{4n}), R_{(2,1,...,1)}(C_{n}) (n ≥ 8) and, R_{(2,2,1,...,1)}(C_{4n}) (n ≥ 12) are graceful.

For any permutation f on 1, …, n, the f-permutation graph on a graph G, P (G, f), consists of two disjoint copies of G, G_{1} and G_{2}, each of which has vertices labeled v_{1}, v_{2}, …, v_{n} with n edges obtained by joining each v_{i} in G_{1} to v_{f(i)} in G_{2}. In 1983 Lee (see [592]) conjectured that for all n > 1 and all permutations on 1, 2, …, n, the permutation graph P (P_{n}, f) is graceful. Lee, Wang, and Kiang [592] proved that P (P_{2k}, f) is graceful when f = (12)(34) ⋯ (k, k + 1) ⋯ (2k − 1, 2k). They conjectured that if G is a graceful nonbipartite graph with n vertices, then for any permutation f on 1, 2, …, n, the permutation graph P (G, f) is graceful. Some families of graceful permutation graphs are given in [538].

Gnanajothi [367, p. 51] calls a graph G bigraceful if both G and its line graph are graceful. She shows the following are bigraceful: P_{m}; P_{m} × P_{n}; C_{n} if and only if n ≡ 0, 3 (mod 4); S_{n}; K_{n} if and only if n ≤ 3; and B_{n} if and only if n ≡ 3 (mod 4). She also shows that K_{m,n} is not bigraceful when n ≡ 3 (mod 4). (Gangopadhyay and Hebbare [351] used the term “bigraceful” to mean a bipartite graceful graph.) Murugan and Arumugan [672] have shown that graphs obtained from C_{4} by attaching two disjoint paths of equal length to two adjacent vertices are bigraceful.

Several well-known isolated graphs have been examined. Graceful labelings have been found for the Petersen graph [334], the cube [355], the icosahedron and the dodecahedron. On the other hand, Graham and Sloane [380] showed that all of these except the cube are harmonious. Winters [957] verified that the Grőtzsch graph (see [190, p. 118]), the Heawood graph (see [190, p. 236]), and the Herschel graph (see [190, p. 53]) are graceful. Graham and Sloane [380] determined all harmonious graphs with at most five vertices. Seoud and Youssef [770] did the same for graphs with six vertices.

Jirimutu, Wang, and Xirong [466] define a digraph G(V, E) to be graceful if there exists an injection f from V (G) to {0, 1, 2, …, |E|} such that the induced function on the set of edges given by f′(u, v) = (f(v) − f(u))(mod |E| + 1) is a bijection. They prove that the directed graph consisting of n copies of a directed m-cycle having a vertex in common is graceful for m = 9, 11, 13 and all even n.

2.8 Summary

The results and conjectures discussed above are summarized in the tables following. The letter G after a class of graphs indicates that the graphs in that class are known to be graceful; a question mark indicates that the gracefulness of the graphs in the class is an open problem; we put a question mark after a “G” if the graphs have been conjectured to be graceful. The analogous notation with the letter H is used to indicate the status of the graphs with regard to being harmonious. The tables impart at a glimpse what has been done and what needs to be done to close out a particular class of graphs. Of course, there is an unlimited number of graphs one could consider. One wishes for some general results that would handle several broad classes at once but the experience of many people suggests that this is unlikely to occur soon. The Graceful Tree Conjecture alone has withstood the efforts of scores of people over the past four decades. Analogous sweeping conjectures are probably true but appear hopelessly difficult to prove.

Table 1: Summary of Graceful Results

Graph  Graceful
trees  G if ≤ 27 vertices [38]

G if symmetrical [163] G if at most 4 end-vertices [448] G? Ringel-Kotzig G caterpillars [733] G? lobsters [159]

cycles C_{n}  G iff n ≡ 0, 3 (mod 4) [733]
wheels W_{n}  G [334], [440]
helms (see §2.2)  G [55]
webs (see §2.2)  G [472]
gears (see §2.2)  G [633]
cycles with P_{k}-chord (see §2.2)  G [274], [632], [501], [711]
C_{n} with k consecutive chords (see §2.2) G if k = 2, 3, n − 3 [491], [500]
unicyclic graphs  G? iff G ≠ C_{n}, n ≡ 1, 2 (mod 4) [918]
C_{n}^{(t)} (see §2.2)  n = 3 G iff t ≡ 0, 1 (mod 4)

[160], [162] G? if nt ≡ 0, 3 (mod 4) [492] G if n = 6, t even [492] G if n = 4, t > 1 [803] G if n = 5, t > 1 [980] G if n = 7 and t ≡ 0, 3 (mod 4) [986] G if n = 9 and t ≡ 0, 3 (mod 4) [987] G if t = 2 n ≢ 1 (mod 4) [713], [188] G if n = 11 [973]

Table 1: Summary of Graceful Results continued

Graph  Graceful
triangular snakes (see §2.2)  G iff number of blocks ≡ 0, 1 (mod 4) [669]
K_{4}-snakes (see §2.2)  ?
quadrilateral snakes (see §2.2)  G [367], [713]
crowns C_{n} ⊙ K_{1}  G [334]
C_{n} ⊙ P_{k}  G [753]
grids P_{m} × P_{n}  G [20]
prisms C_{m} × P_{n}  G if n = 2 [337], [983]

G if m even [449]

G if m odd and 3 ≤ n ≤ 12 [449]

G if m = 3 [833] G if m = 6 see [985] G if m ≡ 2 (mod 4) and n ≡ 3 (mod 4) [985]

K_{m} × P_{n}  G if (m, n) = (4, 2), (4, 3), (4, 4), (4, 5), (5, 2)[855]

not G if (m, n) = (3, 3), (6, 2), (7, 2), (8, 2), (9, 2)(10, 2) [855] not G? for (m, 2) with m > 5 [855]

K_{m,n} ⊙ K_{1}  G [464]
torus grids C_{m} × C_{n}  G if m ≡ 0 (mod 4), n even [468]

not G if m, n odd (parity condition)

vertex-deleted C_{m} × P_{n}  G if n = 2 [350]
edge-deleted C_{m} × P_{n}  G if n = 2 [350]
Möbius ladders M_{n} (see §2.3)  G [345]
stacked books S_{m} × P_{n} (see §2.3) n = 2, G iff m ≢ 3 (mod 4) [640], [273], [349]

G if m even [349]

n-cube K_{2} × K_{2} × ⋯ × K_{2}  G [507]

Graph  Graceful
K_{4} × P_{n}  G if n = 2, 3, 4, 5 [700]
K_{n}  G iff n ≤ 4 [371], [832]
K_{m,n}  G [733], [371]
K_{1,m,n}  G [51]
K_{1,1,m,n}  G [367]
windmills K_{n}^{(m)}(n > 3) (see §2.4)  G if n = 4, m ≤ 22 [449]

G? if n = 4, m ≥ 4 [159] G if n = 4, 4 ≤ m ≤ 22 [449] not G if n = 4, m = 2, 3 [159] not G if (m, n) = (2, 5) [162] not G if n > 5 [500]

B(n, r, m) r > 1 (see §2.4)  G if (n, r) = (3, 2), (4, 3) [493], (4,2) [273]

G (n, r, m) = (5, 2, 2) [855]

not G for (n, 2, 2) for n > 5 [161], [855]

mK_{n} (see §2.5)  G iff m = 1, n ≤ 4 [511]
C_{s} ∪ P_{n}  ? G iff s + n ≥ 7 [338]

G if s = 3 [336], s = 4 [338], s = 5 [490] G if s > 4, n = 2 [353] G if s = 2n + 1 [336] G if s = 2k, n ≥ k + 1 [760]

C_{p} ∪ C_{q}  G iff p + q ≡ 0, 3 (mod 4) [338]

Graph  Graceful
C_{n} ∪ K_{p,q}  for n > 8 G iff n ≡ 0, 3 (mod 4) [997]

G C_{6} × K_{1,2n+1} [136]

G C_{3} × K_{m,n} iff m, n ≥ 2 [769] G C_{4} × K_{m,n} iff (m, n) ≠ (1, 1)[769] G C_{7} × K_{m,n} [769] G C_{8} × K_{m,n} [769]

K_{i} ∪ K_{m,n}  G [136]
⋃_{i=1}^{t} K_{m_{i},n_{i}}  G 2 ≤ m_{i} < n_{i} [136]
C_{m} ∪ ⋃_{i=1}^{t} K_{m_{i},n_{i}}  G 2 ≤ m_{i} < n_{i},

m ≡ 0 or 3 (mod 4), m ≥ 11 [136]

G + \overline{K_{t}}  G for connected G
double cones C_{n} + \overline{K_{2}}  G for n = 3, 4, 5, 7, 8, 9, 11, 12

not G for n ≡ 2 (mod 4)

t-point suspension C_{n} + \overline{K_{t}} G if n ≡ 0 or 3 (mod 12) [171]

not G if t is even and n ≡ 2, 6, 10 (mod 12) G if n = 4, 7, 11 or 19 [171] G if n = 5 or 9 and t = 2 [171]

P_{n}^{2} (see §2.7)  G [537]
Petersen P (n, k) (see §2.7) G for n = 5, 6, 7, 8, 9, 10 [718], (n, k) = (8t, 3)[924]

Table 2: Summary of Harmonious Results

Graph  Harmonious
trees  H if ≤ 26 vertices [38]

H? [380]

H caterpillars [380]

? lobsters

cycles C_{n}  H iff n ≡ 1, 3 (mod 4) [380]
wheels W_{n}  H [380]
helms (see §2.2)  H [367], [617]
webs (see §2.2)  H if cycle is odd
gears (see §2.2)  H [242]
cycles with P_{k}-chord (see §2.2)  ?
C_{n} with k consecutive chords (see §2.2) ?
unicyclic graphs  ?
C_{n}^{(t)} (see §2.2)  n = 3 H iff t ≢ 2 (mod 4) [380]

H if n = 4, t > 1 [803]

triangular snakes (see §2.2)  H if number of blocks is odd [972]

not H if number of blocks ≡ 2 (mod 4) [972]

K_{4}-snakes (see §2.2)  H [378]
quadrilateral snakes (see §2.2)  ?
crowns C_{n} ⊙ K_{1}  H [377], [619]
grids P_{m} × P_{n}  H iff (m, n) ≠ (2, 2) [468]

Table 2: Summary of Harmonious Results continued

Graph  Harmonious
prisms C_{m} × P_{n}  H if n = 2, m ≠ 4 [350]

H if n odd [380]

H if m = 4 and n ≥ 3 [468]

torus grids C_{m} × C_{n},  H if m = 4, n > 1 [468]

not H if m ≢ 0 (mod 4) and n odd [468]

vertex-deleted C_{m} × P_{n}  H if n = 2 [350]
edge-deleted C_{m} × P_{n}  H if n = 2 [350]
Möbius ladders M_{n} (see §2.3)  H iff n ≠ 3 [345]
stacked books S_{m} × P_{n} (see §2.3) n = 2, H if m even [376], [722]

not H m ≡ 3 (mod 4), n = 2,

(parity condition)

H if m ≡ 1 (mod 4), n = 2 [367]

n-cube K_{2} × K_{2} × ⋯ × K_{2}  not H if n = 2, 3 [380]
K_{4} × P_{n}  H [722]
K_{n}  H iff n ≤ 4 [380]
K_{m,n}  H iff m or n = 1 [380]
K_{1,m,n}  H [51]
K_{1,1,m,n}  H [367]
windmills K_{n}^{(m)} (n > 3) (see §2.4) H if n = 4 [444]

m = 2, H? iff n = 4 [380]

not H if m = 2, n odd or 6 [380] not H for some cases m = 3 [608]

Graph  Harmonious
B(n, r, m) r > 1 (see §2.4)  (n, r) = (3, 2), (4, 3) [766]
mK_{n} (see §2.5)  H n = 3, m odd [620]

not H for n odd, m ≡ 2 (mod 4) [620]

nG  H when G is harmonious and n odd [995]
G^{n}  H when G is harmonious and n odd [995]
C_{s} ∪ P_{n}  ?
fans F_{n} = P_{n} + K_{1}  H [380]
nC_{m} + K_{1} n ≢ 0 mod 4  H [242]
double fans P_{n} + \overline{K_{2}}  H [380]
t-point suspension P_{n} + \overline{K_{t}} of P_{n} H [722]
S_{m} + K_{1}  H [367], [230]
t-point suspension C_{n} + \overline{K_{t}} of C_{n} H if n odd and t = 2 [722], [367]

not H if n ≡ 2, 4, 6 (mod 8) and t = 2 [367]

P_{n}^{2} (see §2.7)  H [377], [619]
Petersen P (n, k) (see §2.7)  H [367], [563]

3 Variations of Graceful Labelings

3.1 α-labelings

In 1966 Rosa [733] defined an α-labeling (or α-valuation) as a graceful labeling with the additional property that there exists an integer k so that for each edge xy either f(x) ≤ k < f(y) or f(y) ≤ k < f(x). (Other names for such labelings are balanced and interlaced.) It follows that such a k must be the smaller of the two vertex labels that yield the edge labeled 1. Also, a graph with an α-labeling is necessarily bipartite and therefore can not contain a cycle of odd length. Wu [965] has shown that a necessary condition for a bipartite graph with n edges and degree sequence d_{1}, d_{2}, …, d_{p} to have an α-labeling is that the gcd(d_{1}, d_{2}, …, d_{p}, n) divides n(n − 1)/2.

A common theme in graph labeling papers is to build up graphs that have desired labelings from pieces with particular properties. In these situations, starting with a graph that possesses an α-labeling is a typical approach. (See [230], [377], [241], and [468].) Moreover, Jungreis and Reid [468] showed how sequential labelings of graphs (see Section 4.1) can often be obtained by modifying α-labelings of the graphs.

Graphs with α-labelings have proved to be useful in the development of the theory of graph decompositions. Rosa [733], for instance, has shown that if G is a graph with q edges and has an α-labeling, then for every natural number p, the complete graph K_{2qp+1} can be decomposed into copies of G in such a way that the automorphism group of the decomposition itself contains the cyclic group of order p. In the same vein El-Zanati and Vanden Eynden [300] proved that if G has q edges and admits an α-labeling then K_{qm,qn} can be partitioned into subgraphs isomorphic to G for all positive integers m and n. Although a proof of Ringel’s conjecture that every tree has a graceful labeling has withstood many attempts, examples of trees that do not have α-labelings are easy to construct (see [733]). Kotzig [505] has shown however that almost all trees have αlabelings.

As to which graphs have α-labelings, Rosa [733] observed that the n-cycle has an αlabeling if and only if n ≡ 0 (mod 4) whereas P_{n} always has an α-labeling. Other familiar graphs that have α-labelings include caterpillars [733], the n-cube [504], B_{4n+1} (i.e., books with 4n+1 pages) [349], C_{2m}∪C_{2m} and C_{4m}∪C_{4m}∪C_{4m} for all m > 1 [506], C_{4m}∪C_{4m}∪C_{4n} for all (m, n) ≠ 1, 1) [314], P_{n} × Q_{n} [640], K_{1,2k} × Q_{n} [640], C_{4m} ∪ C_{4m} ∪ C_{4m} ∪ C_{4m} [527], C_{4m} ∪ C_{4n+2} ∪ C_{4r+2}, C_{4m} ∪ C_{4n} ∪ C_{4r} when m + n ≤ r [10], C_{4m} ∪ C_{4n} ∪ C_{4r} ∪ C_{4s} when m ≥ n+r+s [6], C_{4m} ∪C_{4n} ∪C_{4r+2} ∪C_{4s+2} when m ≥ n+r+s+1 [6], ((m+1)^{2} +1)C_{4} for all m [1010], k^{2}C_{4} for all k [1010], and (k^{2} + k)C_{4} for all k [1010]. Abrham and Kotzig [8] have shown that kC_{4} has an α-labeling for 1 ≤ k ≤ 10 and that if kC_{4} has an α-labeling then so does (4k + 1)C_{4}, (5k + 1)C_{4} and (9k + 1)C_{4}. Eshghi [309] proved that 5C_{4k} has an α-labeling for all k. In [314] Eshghi and Carter show several families of graphs of the form C_{4n_{1}} ∪ C_{4n_{2}} ∪ ⋯ ∪ C_{4n_{k}} have α-labelings.

Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] have shown that if m ≡ 0 (mod 4) then the one-point union of 2, 3, or 4 copies of C_{m} admits an α-labeling, and if m ≡ 2 (mod 4) then the one-point union of 2 or 4 copies of C_{m} admits an α-labeling.

They conjecture that the one-point union of n copies of C_{m} admits an α-labeling if and only if mn ≡ 0 (mod 4).

In his 2001 Ph. D. thesis Selvaraju [754] investigated the one-point union of complete bipartite graphs. He proves that the one-point unions of the following forms have an α-labeling: K_{m,n_{1}} and K_{m,n_{2}}; K_{m_{1},n_{1}}, K_{m_{2},n_{2}}, and K_{m_{3},n_{3}} where m_{1} ≤ m_{2} ≤ m_{3} and n_{1} < n_{2} < n_{3}; K_{m_{1},n}, K_{m_{2},n}, and K_{m_{3},n} where m_{1} < m_{2} < m_{3} ≤ 2n.

Zhile [1010] uses C_{m}(n) to denote the connected graph all of whose blocks are C_{m} and whose block-cutpoint-graph is a path. He proves that for all positive integers m and n, C_{4m}(n) has an α-labeling but C_{m}(n) does not have an α-labeling when m is odd.

Abrham and Kotzig [10] have proved that C_{m} ∪ C_{n} has an α-labeling if and only if both m and n are even and m + n ≡ 0 (mod 4). Kotzig [506] has also shown that C_{4} ∪ C_{4} ∪ C_{4} does not have an α-labeling. He asked if n = 3 is the only integer such that the disjoint union of n copies of C_{4} does not have an α-labeling. This was confirmed by Abrham and Kotzig in [9]. Eshghi [308] proved that every 2-regular bipartite graph with 3 components has an α-labeling if and only if the number of edges is a multiple of four except for C_{4} ∪ C_{4} ∪ C_{4}. In [311] Eshghi gives more results on the existence of α-labelings for various families of disjoint union of cycles.

Jungreis and Reid [468] investigated the existence of α-labelings for graphs of the form P_{m} × P_{n}, C_{m} × P_{n}, and C_{m} × C_{n} (see also [348]). Of course, the cases involving C_{m} with m odd are not bipartite, so there is no α-labeling. The only unresolved cases among these three families are C_{4m+2} × P_{2n+1} and C_{4m+2} × C_{4n+2}. All other cases result in α-labelings. Balakrishman [119] uses the notation Q_{n}(G) to denote the graph P_{2} × P_{2} × ⋯ × P_{2} × G where P_{2} occurs n − 1 times. Snevily [859] has shown that the graphs Q_{n}(C_{4m}) and the cycles C_{4m} with the path P_{n} adjoined at each vertex have α-labelings. He [860] also has shown that compositions of the form G[\overline{K_{n}}] have an α-labeling whenever G does (see §2.3 for the definition of composition). Balakrishman and Kumar [121] have shown that all graphs of the form Q_{n}(G) where G is K_{3,3}, K_{4,4}, or P_{m} have an α-labeling. Balakrishman [119] poses the following two problems. For which graphs G does Q_{n}(G) have an αlabeling? For which graphs G does Q_{n}(G) have a graceful labeling?

Rosa [733] has shown that K_{m,n} has an α-labeling (see also [134]). Barrientos [134] has shown that for n even the graph obtained from the wheel W_{n} by attaching a pendant edge at each vertex has an α-labeling. In [140] Barrientos shows how to construct graceful graphs that are formed from the one-point union of a tree that has an α-labeling, P_{2}, and the cycle C_{n}. In some cases, P_{2} is not needed. Qian [713] has proved that quadrilateral snakes have α-labelings. Fu and Wu [340] showed that if T is a tree that has an αlabeling with partite sets V_{1} and V_{2} then the graph obtained from T by joining new vertices w_{1}, w_{2}, …, w_{k} to every vertex of V_{1} has an α-labeling. Similarly, they prove that the graph obtained from T by joining new vertices w_{1}, w_{2}, …, w_{k} to the vertices of V_{1} and new vertices u_{1}, u_{2}, …, u_{t} to every vertex of V_{2} has an α-labeling. They also prove that if one of the new vertices of either of these two graphs is replaced by a star and every vertex of the star is joined to the vertices of V_{1} or the vertices of both V_{1} and V_{2}, the resulting graphs have α-labelings. Fu and Wu [340] further show that if T is a tree with an α-labeling and the sizes of the two partite sets of T differ at by at most 1, then T ×P_{m} has an α-labeling.

Lee and Liu [543] investigated the mirror graph M(m, n) of K_{m,n} (see §2.3 for the definition) for α-labelings. They proved: M(m, n) has an α-labeling when n is odd or m is even; M(1, n) has an α-labeling when n ≡ 0 (mod 4); M(m, n) does not have an α-labeling when m is odd and n ≡ 2 (mod 4), or when m ≡ 3 (mod 4) and n ≡ 4 (mod 8).

Barrientos [135] defines a chain graph as one with blocks B_{1}, B_{2}, …, B_{m} such that for every i, B_{i} and B_{i+1} have a common vertex in such a way that the block-cutpoint graph is a path. He shows that if B_{1}, B_{2}, …, B_{m} are blocks that have α-labelings then there exists a chain graph G with blocks B_{1}, B_{2}, …, B_{m} that has an α-labeling. He also shows that if B_{1}, B_{2}, …, B_{m} are complete bipartite graphs, then any chain graph G obtained by concatenation of these blocks has an α-labeling.

Wu ([964] and [966]) has given a number of methods for constructing larger graceful graphs from graceful graphs. Let G_{1}, G_{2}, …, G_{p} be disjoint connected graphs. Let w_{i} be in G_{i} for 1 ≤ i ≤ p. Let w be a new vertex not in any G_{i}. Form a new graph ⊕_{w}(G_{1}, G_{2}, …, G_{p}) by adjoining to the graph G_{1} ∪ G_{2} ∪ ⋯ ∪ G_{p} the edges ww_{1}, ww_{2}, …, ww_{p}. In the case where each of G_{1}, G_{2}, …, G_{p} is isomorphic to a graph G that has an α-labeling and each w_{i} is the isomorphic image of the same vertex in G_{i}, Wu shows that the resulting graph is graceful. If f is an α-labeling of a graph, the integer k with the property that for any edge uv either f(u) ≤ k < f(v) or f(v) ≤ k < f(u) is called the boundary value or critical number of f. Wu [964] has also shown that if G_{1}, G_{2}, …, G_{p} are graphs of the same order and have α-labelings where the labelings for each pair of graphs G_{i} and G_{p−i+1} have the same boundary value for 1 ≤ i ≤ n/2, then ⊕_{w}(G_{1}, G_{2}, …, G_{p}) is graceful. In [962] Wu proves that if G has n edges and n + 1 vertices and G has an α-labeling with boundary value λ, where |n − 2λ − 1| ≤ 1, then G × P_{m} is graceful for all m.

Snevily [860] says that a graph G eventually has an α-labeling provided that there is a graph H, called a host of G, which has an α-labeling and that the edge set of H can be partitioned into subgraphs isomorphic to G. He defines the α-labeling number of G to be G_{α} = min{t : there is a host H of G with |E(H)| = t|G|}. Snevily proved that even cycles have α-labeling number at most 2 and he conjectured that every bipartite graph has an α-labeling number. This conjecture was proved by El-Zanati, Fu, and Shiue [299]. There are no known examples of a graph G with G_{α} > 2. In [860] Snevily conjectured that the α-labeling number for a tree with n edges is at most n. Shiue and Fu [827] proved that α-labeling number for a tree with n edges and radius r is at most ⌈r/2⌉n. They also prove that a tree with n edge and radius r decomposes K_{t} for some t ≤ (r + 1)n^{2} + 1.

Given two bipartite graphs G_{1} and G_{2} with partite sets H_{1} and L_{1} and H_{2} and L_{2}, respectively, Snevily [859] defines their weak tensor product G_{1}\overline{⨂}G_{2} as the bipartite graph with vertex set (H_{1} × H_{2}, L_{1} × L_{2}) and with edge (h_{1}, h_{2})(l_{1}, l_{2}) if h_{1}l_{1} ∈ E(G_{1}) and h_{2}l_{2} ∈ E(G_{2}). He proves that if G_{1} and G_{2} have α-labelings then so does G_{1}\overline{⨂}G_{2}. This result considerably enlarges the class of graphs known to have α-labelings.

The sequential join of graphs G_{1}, G_{2}, …, G_{n} is formed from G_{1} ∪ G_{2} ∪ ⋯ ∪ G_{n} by adding edges joining each vertex of G_{i} with each vertex of G_{i+1} for 1 ≤ i ≤ n − 1. Lee and Wang [581] have shown that for all n ≥ 2 and any positive integers a_{1}, a_{2}, …, a_{n} the sequential join of the graphs \overline{K}_{a_{1}}, \overline{K}_{a_{2}}, …, \overline{K}_{a_{n}} has an α-labeling.

In [346] Gallian and Ropp conjectured that every graph obtained by adding a single pendant edge to one or more vertices of a cycle is graceful. Qian [713] proved this conjecture and in the case that the cycle is even he shows the graphs have an α-labeling. He further proves that for n even any graph obtained from an n-cycle by adding one or more pendant edges at some vertices has an α-labeling as long as at least one vertex has degree 3 and one vertex has degree 2.

For any tree T (V, E) whose vertices are properly 2-colored Rosa and Širáň [736] define a bipartite labeling of T as a bijection f : V → {0, 1, 2, …, |E|} for which there is a k such that whenever f(u) ≤ k ≤ f(v), then u and v have different colors. They define the α-size of a tree T as the maximum number of distinct values of the induced edge labels |f(u)−f(v)|, uv ∈ E, taken over all bipartite labelings f of T. They prove that the α-size of any tree with n edges is at least 5(n + 1)/7 and that there exist trees whose α-size is at most (5n + 9)/6. They conjectured that minimum of the α-sizes over all trees with n edges is asymptotically 5n/6. This conjecture has been proved for trees of maximum degree 3 by Bonnington and Širáň [207]. Heinrich and Hell [433] defined the gracesize of a graph G with n vertices as the maximum, over all bijections f: V (G) → {1, 2, …, n}, of the number of distinct values |f(u) − f(v)| over all edges uv of G. So, from Rosa and Širáň’s result, the gracesize of any tree with n edges is at least 5(n + 1)/7.

In [350] Gallian weakened the condition for an α-labeling somewhat by defining a weakly α-labeling as a graceful labeling for which there is an integer k so that for each edge xy either f(x) ≤ k ≤ f(y) or f(y) ≤ k ≤ f(x). Unlike α-labelings, this condition allows the graph to have an odd cycle, but still places a severe restriction on the structure of the graph; namely, that the vertex with the label k must be on every odd cycle. Gallian, Prout, and Winters [350] showed that the prisms C_{n} × P_{2} with a vertex deleted have αlabelings. The same paper reveals that C_{n} × P_{2} with an edge deleted from a cycle has an α-labeling when n is even and a weakly α-labeling when n > 3.

A special case of α-labeling called strongly graceful was introduced by Maheo [640] in 1980. A graceful labeling f of a graph G is called strongly graceful if G is bipartite with two partite sets A and B of the same order s, the number of edges is 2t + s, there is an integer k with t − s ≤ k ≤ t + s − 1 such that if a ∈ A, f(a) ≤ k, and if b ∈ B, f(b) > k, and there is an involution π that is an automorphism of G such that: π exchanges A and B and the s edges aπ(a) where a ∈ A have as labels the integers between t + 1 and t + s. Maheo’s main result is that if G is strongly graceful then so is G × Q_{n}. In particular, she proved that (P_{n} × Q_{n}) × K_{2}, B_{2n}, and B_{2n} × Q_{n} have strongly graceful labelings.

El-Zanati and Vanden Eynden [301] call a strongly graceful labeling a strong α-labeling They show that if G has a strong α-labeling, then G × P_{n} has an α-labeling. They show that K_{m,2} × K_{2} has a strong α-labeling and that K_{m,2} × P_{n} has an α-labeling. They also show that if G is a bipartite graph with one more vertex than the number of edges, and if G has an α-labeling such that the cardinalities of the sets of the corresponding bipartition of the vertices differ by at most 1, then G × K_{2} has a strong α-labeling and G × P_{n} has an α-labeling. El-Zanati and Vanden Eynden [301] also note that K_{3,3} × K_{2}, K_{3,4} × K_{2}, K_{4,4}×K_{2}, and C_{4k} ×K_{2} all have strong α-labelings. El-Zanati and Vanden Eynden proved that K_{m,2} × Q_{n} has a strong α-labeling and that K_{m,2} × P_{n} has an α-labeling for all n. They also prove that if G is a connected bipartite graph with partite sets of odd order such that in each partite set each vertex has the same degree, then G × K_{2} does not have a strong α-labeling. As a corollary they have that K_{m,n} × K_{2} does not have a strong α-labeling when m and n are odd.

An α-labeling f of a graph G is called free by El-Zanati and Vanden Eynden in [302] if the critical number k (in the definition of α-labeling) is greater than 2 and if neither 1 nor k − 1 is used in the labeling. Their main result is that the union of graphs with free α-labelings has an α-labeling. In particular, they show that K_{m,n}, m > 1, n > 2, has a free α-labeling. They also show that Q_{n}, n ≥ 3, and K_{m,2} × Q_{n}, m > 1, n ≥ 1, have free α-labelings. El-Zanati [personal communication] has shown that the Heawood graph has a free α-labeling.

For connected bipartite graphs Grannell, Griggs, and Holroyd [381] introduced a labeling that lies between α-labelings and graceful labelings. They call a vertex labeling f of a bipartite graph G with q edges and partite sets D and U gracious if f is a bijection from the vertex set of G to {0, 1, …, q} such that the set of edge labels induced by f(u) − f(v) for every edge uv with u ∈ U and v ∈ D is {1, 2, …, q}. Thus a gracious labeling of G with partite sets D and U is a graceful labeling in which every vertex in D has a label lower than every adjacent vertex. They verified by computer that every tree of size up to 20 has a gracious labeling. This led them to conjecture that every tree has a gracious labeling. For any k > 1 and any tree T Grannell et al. say that T has a gracious k-labeling if the verticies of T can be partitioned into sets D and U in such a way that there is a function f from the verticies of G to the integers modulo k such that the edge labels induced by f(u) − f(v) where u ∈ U and v ∈ D have the following properties: the number of edges labeled with 0 is one less than the number of verticies labeled with 0 and for each nonzero integer t the number of edges labeled with t is the same as the number of verticies labeled with t. They prove that every nontrivial tree has a k-gracious labeling for k = 2, 3, 4, and 5 and that caterpillars are k-gracious for all k ≥ 2.

The same labeling that is called gracious by Grannell, Griggs, and Holroyd is called a near α-labeling by El-Zanati, Kenig, and Vanden Eynden [303]. The latter prove that if G is a graph with n edges that has a near α-labeling then there exists a cyclic Gdecomposition of K_{2nx+1} for all positive integers x and a cyclic G-decomposition of K_{n,n}. They further prove that if G and H have near α-labelings, then so does their weak tensor product (see earlier part of this section) with respect to the corresponding vertex partitions. They conjecture that every tree has a near α-labeling.

Another kind of labelings for trees was introduced by Ringel, Llado, and Serra [725] in an approach to proving their conjecture K_{n,n} is edge-decomposable into n copies of any given tree with n edges. If T is a tree with n edges and partite sets A and B, they define a labeling f from the set of vertices to {1, 2, …, n} to be a bigraceful labeling of T if f restricted to A is injective, f restricted to B is injective, and the edge labels given by f(y) − f(x) where yx is an edge with y in B and x in A is the set {0, 1, 2, …, n − 1}. (Notice that this terminology conflicts with that given in Section 2.7 In particular, the Ringel, Llado, and Serra bigraceful does not imply the usual graceful.) Among the graphs that they show are bigraceful are: lobsters, trees of diameter at most 5, stars S_{k,m} with k spokes of paths of length m, and complete d-ary trees for d odd. They also prove that if T is a tree then there is a vertex v and a nonnegative integer m such that the addition of m leaves to v results in a bigraceful tree. They conjecture that all trees are bigraceful.

Table 3 summarizes some of the main results about α-labelings. α indicates that the graphs have an α-labeling.

Table 3: Summary of Results on α-labelings

Graph  α-labeling
cycles C_{n}  α iff n ≡ 0 (mod 4) [733]
caterpillars  α [733]
n-cube  α [504]
books B_{2n}, B_{4n+1}  α [640],[349]
C_{m} ∪ C_{n}  α iff m, n are even and m + n ≡ 0 (mod 4)[10]
C_{4m} ∪ C_{4m} ∪ C_{4m} (m > 1) α [506]
C_{4m} ∪ C_{4m} ∪ C_{4m} ∪ C_{4m}  α [506]
P_{n} × Q_{n}  α [640]
B_{2n} × Q_{n}  α [640]
K_{1,n} × Q_{n}  α [640]
K_{m,2} × Q_{n}  α [301]
K_{m,2} × P_{n}  α [301]
P_{2} × P_{2} × ⋯ × P_{2} × G  α when G = C_{4m}, P_{m}, K_{3,3}, K_{4,4} [859]
P_{2} × P_{2} × ⋯ × P_{2} × P_{m}  α [859]
P_{2} × P_{2} × ⋯ × P_{2} × K_{m,m} α [859] when m = 3 or 4
G[\overline{K_{n}}]  α when G is α [860]

3.2 k-graceful Labelings

A natural generalization of graceful graphs is the notion of k-graceful graphs introduced independently by Slater [850] in 1982 and by Maheo and Thuillier [641] in 1982. A graph G with q edges is k-graceful if there is labeling f from the vertices of G to {0, 1, 2, …, q+k−1} such that the set of edge labels induced by the absolute value of the difference of the labels of adjacent vertices is {k, k + 1, …, q + k − 1}. Obviously, 1-graceful is graceful and it is readily shown that any graph that has an α-labeling is k-graceful for all k. Graphs that are k-graceful for all k are sometimes called arbitrarily graceful. Ng [680] has shown that there are graphs that are k-graceful for all k but do not have an α-labeling.

Results of Maheo and Thuillier [641] together with those of Slater [850] show that: C_{n} is k-graceful if and only if either n ≡ 0 or 1 (mod 4) with k even and k ≤ (n − 1)/2, or n ≡ 3 (mod 4) with k odd and k ≤ (n^{2} − 1)/2. Maheo and Thuillier [641] also proved that the wheel W_{2k+1} is k-graceful and conjectured that W_{2k} is k-graceful when k ≠ 3 or k ≠ 4. This conjecture was proved by Liang, Sun, and Xu [598]. Kang [470] proved that P_{m} × C_{4n} is k-graceful for all k. Lee and Wang [579] showed that the graphs obtained from a nontrivial path of even length by joining every other vertex to one isolated vertex (a lotus), the graphs obtained from a nontrivial path of even length by joining every other vertex to two isolated vertices (a diamond), and the graphs obtained by arranging vertices into a finite number of rows with i vertices in the ith row and in every row the jth vertex in that row is joined to the jth vertex and j + 1st vertex of the next row (a pyramid) are k-graceful. Liang and Liu [595] have shown that K_{m,n} is k-graceful. Bu, Gao, and Zhang [199] have proved that P_{n} × P_{2} and (P_{n} × P_{2}) ∪ (P_{n} × P_{2}) are k-graceful for all k. Acharya (see [14]) has shown that a k-graceful Eulerian graph with q edges must satisfy one of the following conditions: q ≡ 0 (mod 4), q ≡ 1 (mod 4) if k is even, or q ≡ 3 (mod 4) if k is odd. Bu, Zhang, and He [204] have shown that an even cycle with a fixed number of pendant edges adjoined to each vertex is k-graceful. Lu, Pan, and Li [630] have proved that K_{1,m} ∪ K_{p,q} is k-graceful when k > 1, and p and q are at least 2. Jirimutu, Bao, and Kong [465] have shown that the graphs obtained from K_{2,n} (n ≥ 2) and K_{3,n} (n ≥ 3) by attaching r ≥ 2 edges at each vertex is k-graceful for all k ≥ 2. Seoud and Elsakhawi [763] proved: paths and ladders are arbitrarily graceful; and for n ≥ 3, K_{n} is k-graceful if and only if k = 1 and n = 3 or 4.

Yao, Cheng, Zhongfu, and Yao [989] have shown: a tree of order p with maximum degree at least p/2 is k-graceful for some k; if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, then T is k-graceful for some positive k; if a tree T has two edges u_{1}u_{2} and u_{2}u_{3} such that the three components T_{1}, T_{2}, and T_{3} of T − {u_{1}u_{2}, u_{2}u_{3}} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2, d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, and d_{T_{3}}(u_{3}) ≥ |T_{3}|/2, then T is k-graceful for some k > 1; and every Skolem-graceful (see §3.4 for the definition) tree is k-graceful for all k ≥ 1. They conjecture that every tree is k-graceful for some k > 1.

Several authors have investigated the k-gracefulness of various classes of subgraphs of grid graphs. Acharya [12] proved that all 2-dimensional polyminoes that are convex and Eulerian are k-graceful for all k; Lee [531] showed that Mongolian tents and Mongolian villages are k-graceful for all k (see §2.3 for the definitions); Lee and K. C. Ng [547] proved that all Young tableaus (see §2.3 for the definitions) are k-graceful for all k. (A special case of this is P_{n} × P_{2}.) Lee and H. K. Ng [547] subsequently generalized these results on Young tableaus to a wider class of planar graphs.

Duan and Qi [293] use G_{t}(m_{1}, n_{1}; m_{2}, n_{2}; …; m_{s}, n_{s}) to denote the graph composed of the s complete bipartite graphs K_{m_{1},n_{1}}, K_{m_{2},n_{2}}, …, K_{m_{s},n_{s}} that have only t (1 ≤ t ≤ min{m_{1}, m_{2}, …, m_{s}}) common vertices but no common edge and G(m_{1}, n_{1}; m_{2}, n_{2}) to denote the graph composed of the complete bipartite graphs K_{m_{1},n_{1}}, K_{m_{2},n_{2}} with exactly one common edge. They prove that these graphs are k-graceful graphs for all k.

Let c, m, p_{1}, p_{2}, …, p_{m} be positive integers. For i = 1, 2, …, m, let S_{i} be a set of p_{i} + 1 integers and let D_{i} be the set of positive differences of the pairs of elements of S_{i}. If all these differences are distinct then the system D_{1}, D_{2}, …, D_{m} is called a perfect system of difference sets starting at c if the union of all the sets D_{i} is c, c + 1, …, c − 1 +

∑_{i=1}^{m} ( \begin{matrix}p_{i} + 1 \\ 2 \end{matrix} ). There is a relationship between k-graceful graphs and perfect systems

of difference sets. A perfect system of difference sets starting with c describes a c-graceful labeling of a graph that is decomposable into complete subgraphs. A survey of perfect systems of difference sets is given in [4].

Acharya and Hegde [23] generalized k-graceful labelings to (k, d)-graceful labelings by permitting the vertex labels to belong to {0, 1, 2, …, k + (q − 1)d} and requiring the set of edge labels induced by the absolute value of the difference of labels of adjacent vertices to be {k, k + d, k + 2d, …, k + (q − 1)d}. They also introduce an analog of α-labelings in the obvious way. Notice that a (1,1)-graceful labeling is a graceful labeling and a (k, 1)-graceful labeling is a k-graceful labeling. Bu and Zhang [203] have shown: K_{m,n} is (k, d)-graceful for all k and d; for n > 2, K_{n} is (k, d)-graceful if and only if k = d and n ≤ 4; if m_{i}, n_{i} ≥ 2 and max{m_{i}, n_{i}} ≥ 3, then K_{m_{1},n_{1}} ∪ K_{m_{2},n_{2}} ∪ ⋯ ∪ K_{m_{r},n_{r}} is (k, d)-graceful for all k, d, and r; if G has an α-labeling, then G is (k, d)-graceful for all k and d; a k-graceful graph is a (kd, d)-graceful graph; a (kd, d)-graceful connected graph is k-graceful; and a (k, d)-graceful graph with q edges that is not bipartite must have k ≤ (q − 2)d.

Let T be a tree with adjacent vertices u_{0} and v_{0} and pendant vertices u and v such that the length of the path u_{0} − u is the same as the length of the path v_{0} − v. Hegde and Shetty [429] call the graph obtained from T by deleting u_{0}v_{0} and joining u and v an elementary parallel transformation of T. They say that a tree T is a T_{p}-tree if it can be transformed into a path by a sequence of elementary parallel transformations. They prove that every T_{p}-tree is (k, d)-graceful for all k and d and every graph obtained from a T_{p}-tree by subdividing each edge of the tree is (k, d)-graceful for all k and d.

Yao, Cheng, Zhongfu, and Yao [989] have shown: a tree of order p with maximum degree at least p/2 is (k, d)-graceful for some k and d; if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and T_{2} is a caterpillar, then T is Skolem-graceful (see §3.4 for the definition); if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, then T is (k, d)-graceful for some k > 1 and d > 1; if a tree T has two edges u_{1}u_{2} and u_{2}u_{3} such that the three components T_{1}, T_{2}, and T_{3} of T − {u_{1}u_{2}, u_{2}u_{3}} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2, d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, and d_{T_{3}}(u_{3}) ≥ |T_{3}|/2, then T is (k, d)-graceful for some k > 1 and d > 1; and every Skolem-graceful tree is (k, d)-graceful for k ≥ 1 and d > 0. They conjecture that every tree is (k, d)-graceful for some k > 1 and d > 1.

Hegde [419] has proved the following: if a graph is (k, d)-graceful for odd k and even d, then the graph is bipartite; if a graph is (k, d)-graceful and contains C_{2j+1} as a subgraph, then k ≤ jd(q − j − 1); K_{n} is (k, d)-graceful if and only if n ≤ 4; C_{4t} is (k, d)-graceful for all k and d; C_{4t+1} is (2t, 1)-graceful; C_{4t+2} is (2t − 1, 2)-graceful; and C_{4t+3} is (2t + 1, 1)- graceful.

Hegde [417] calls a (k, d)-graceful graph (k, d)-balanced if it has a (k, d)-graceful labeling f with the property that there is some integer m such that for every edge uv either f(u) ≤ m and f(v) > m, or f(u) > m and f(v) ≤ m. He proves that if a graph is (1, 1)- balanced then it is (k, d)-graceful for all k and d and that a graph is (1, 1)-balanced graph if and only if it is (k, k)-balanced for all k. He conjectures that all trees are (k, d)-balanced for some values of k and d.

Slater [853] has extended the definition of k-graceful graphs to countable infinite graphs in a natural way. He proved that all countably infinite trees, the complete graph with countably many vertices, and the countably infinite Dutch windmill is k-graceful for all k.

More specialized results on k-graceful labelings can be found in [531], [547], [550], [850], [198], [200], [199], and [240].

3.3 γ-Labelings

In 2004 Chartrand, Erwin, VanderJagt, and Zhang [231] define a γ-labeling of a graph G of size m as a one-to-one function f from the vertices of G to {0, 1, 2, …, m} that induces an edge labeling f′ defined by f′(uv) = |f(u) − f(v)| for each edge uv. They define the following parameters of a γ-labeling: val(f) = Σf′(e) over all edges e of G; valmax(G) = max{val(f) : f is a γ-labeling of G}, valmin(G) = min{val(f) : f is a γ − labeling of G}. Among their results are the following: valmin(P_{n}) = valmax(P_{n}) = ⌊(n^{2} − 2)/2⌋; valmin(C_{n}) = 2(n − 1); for n ≥ 4, n even, valmax(C_{n}) = n(n+2)/2; for n ≥ 3, n odd, valmax(C_{n}) = (n−1)(n+3)/2; valmin(K_{n}) =

( \begin{matrix}n + 1 \\ 3 \end{matrix} ); for odd n, valmax(K_{n}) = (n^{2}−1)(3n^{2}−5n+6)/24; for even n, valmax(K_{n}) =

n(3n^{3} − 5n^{2} + 6n − 4)/24; for every n ≥ 3, valmin(K_{1,n−1}) = ( \begin{matrix}⌊\frac{n+1}{2}⌋ \\ 2 \end{matrix} ) + ( \begin{matrix}⌈\frac{n+1}{2}⌉ \\ 2 \end{matrix} );

valmax(K_{1,n−1}) = ( \begin{matrix}n \\ 2 \end{matrix} ); for a connected graph of order n and size m, valmin(G) = m

if and only if G is isomorphic to P_{n}; if G is maximal outerplanar of order n ≥ 2, valmin(G) ≥ 3n−5 and equality occurs if and only if G = P_{n}^{2}; if G is a connected r-regular bipartite graph of order n and size m where r ≥ 2, then valmax(G) = rn(2m − n + 2)/4.

In another paper on γ-labelings of trees Chartrand, Erwin, VanderJagt, and Zhang [232] prove for p, q ≥ 2, valmin(S_{p,q}) (that is, the graph obtained by joining the centers of K_{1,p} and K_{1,q} by an edge)= (⌊p/2⌋+1)^{2}+(⌊q/2⌋+1)^{2}−(n_{p}⌊p/2⌋+1)^{2}+(n_{q}⌊(q+2)/2⌋+1)^{2}), where n_{i} is 1 if i is even and n_{i} is 0 if n_{i} is odd; valmin(S_{p,q}) = (p^{2}+q^{2}+4pq−3p−3q+2)/2; for a connected graph G of order n at least 4, valmin(G) = n if and only if G is a caterpillar with maximum degree 3 and has a unique vertex of degree 3; for a tree T of order n at least 4, maximum degree Δ, and diameter d, valmin(T) ≥ (8n + Δ^{2} − 6Δ − 4d + δ_{Δ})/4 where δ_{Δ} is 0 if Δ is even and δ_{Δ} is 0 if Δ is odd. They also give a characterization of all trees of order n at least 5 whose minimum value is n + 1.

3.4 Skolem-Graceful Labelings

A number of authors have invented analogues of graceful graphs by modifying the permissible vertex labels. For instance, Lee (see [572]) calls a graph G with p vertices and q edges Skolem-graceful if there is an injection from the set of vertices of G to {1, 2, …, p} such that the edge labels induced by |f(x) − f(y)| for each edge xy are 1, 2, …, q. A necessary condition for a graph to be Skolem-graceful is that p ≥ q + 1. Lee and Wui [593] have shown that a connected graph is Skolem-graceful if and only if it is a graceful tree. Yao, Cheng, Zhongfu, and Yao [989] have shown that a tree of order p with maximum degree at least p/2 is Skolem-graceful. Although the disjoint union of trees cannot be graceful, they can be Skolem-graceful. Lee and Wui [593] prove that the disjoint union of 2 or 3 stars is Skolem-graceful if and only if at least one star has even size. In [257] Choudum and Kishore show that the disjoint union of k copies of the star K_{1,2p} is Skolem graceful if k ≤ 4p + 1 and the disjoint union of any number of copies of K_{1,2} is Skolem graceful. For k ≥ 2, let St(n_{1}, n_{2}, …, n_{k}) denote the disjoint union of k stars with n_{1}, n_{2}, …, n_{k} edges. Lee, Wang, and Wui [586] showed that the 4-star St(n_{1}, n_{2}, n_{3}, n_{4}) is Skolem-graceful for some special cases and conjectured that all 4-stars are Skolem-graceful. Denham, Leu, and Liu [276] proved this conjecture. Kishore [490] has shown that a necessary condition for St(n_{1}, n_{2}, …, n_{k}) to be Skolem graceful is that some n_{i} is even or k ≡ 0 or 1 (mod 4). He conjectures that each one of these conditions is sufficient. Choudum and Kishore [255] proved that all 5-stars are Skolem graceful.

Lee, Quach, and Wang [558] showed that the disjoint union of the path P_{n} and the star of size m is Skolem-graceful if and only if n = 2 and m is even or n ≥ 3 and m ≥ 1. It follows from the work of Skolem [845] that nP_{2}, the disjoint union of n copies of P_{2}, is Skolem-graceful if and only if n ≡ 0 or 1 (mod 4). Harary and Hsu [402] studied Skolemgraceful graphs under the name node-graceful. Frucht [336] has shown that P_{m} ∪ P_{n} is Skolem-graceful when m + n ≥ 5. Bhat-Nayak and Deshmukh [167] have shown that P_{n_{1}} ∪ P_{n_{2}} ∪ P_{n_{3}} is Skolem-graceful when n_{1} < n_{2} ≤ n_{3}, n_{2} = t(n_{1} + 2) + 1 and n_{1} is even and when n_{1} < n_{2} ≤ n_{3}, n_{2} = t(n_{1} + 3) + 1 and n_{1} is odd. They also prove that the graphs of the form P_{n_{1}} ∪ P_{n_{2}} ∪ ⋯ ∪ P_{n_{i}} where i ≥ 4 are Skolem-graceful under certain conditions. In [279] Deshmukh states the following results: the sum of all the edges on any cycle in a Skolem graceful graph is even ; C_{5} ∪K_{1,n} if and only if n = 1 or 2; C_{6} ∪K_{1,n} if and only if n = 2 or 4.

Youssef [994] proved that if G is Skolem-graceful, then G + \overline{K_{n}} is graceful. In [998] Youssef shows that that for all n ≥ 2, P_{n} ∪ S_{m} is Skolem-graceful if and only if n ≥ 3 or n = 2 and m is even. Yao, Cheng, Zhongfu, and Yao [989] have shown that if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and T_{2} is a caterpillar, then T is Skolem-graceful.

Yao, Cheng, Zhongfu, and Yao [989] have shown that if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, then T is Skolem-graceful.

Mendelsohn and Shalaby [651] defined a Skolem labeled graph G(V, E) as one for which there is a positive integer d and a function L: V → {d, d + 1, …, d + m}, satisfying (a) there are exactly two vertices in V such that L(v) = d + i, 0 ≤ i ≤ m; (b) the distance in G between any two vertices with the same label is the value of the label; and (c) if G′ is a proper spanning subgraph of G, then L restricted to G′ is not a Skolem labeled graph. Note that this definition is different from the Skolem-graceful labeling of Lee, Quach, and Wang. Mendelsohn [650] established the following: any tree can be embedded in a Skolem labeled tree with O(v) vertices; any graph can be embedded as an induced subgraph in a Skolem labeled graph on O(v^{3}) vertices; for d = 1, there is a Skolem labeling or the minimum hooked Skolem (with as few unlabeled vertices as possible) labeling for paths and cycles; for d = 1, there is a minimum Skolem labeled graph containing a path or a cycle of length n as induced subgraph. In [650] Mendelsohn and Shalaby prove that the necessary conditions in [651] are sufficient for a Skolem or minimum hooked Skolem labeling of all trees consisting of edge-disjoint paths of the same length from some fixed vertex. Graham, Pike, and Shalaby [379] obtained various Skolem labeling results for grid graphs. Among them are P_{1} × P_{n} and P_{2} × P_{n} have Skolem labelings if and only if n ≡ 0 or 1 mod 4; and P_{m} × P_{n} has a Skolem labeling for all m and n at least 3.

3.5 Odd-Graceful Labelings

Gnanajothi [367, p. 182] defined a graph G with q edges to be odd-graceful if there is an injection f from V (G) to {0, 1, 2, …, 2q − 1} such that, when each edge xy is assigned the label |f(x) − f(y)|, the resulting edge labels are {1, 3, 5, …, 2q − 1}. She proved that the class of odd-graceful graphs lies between the class of graphs with α-labelings and the class of bipartite graphs by showing that every graph with an α-labeling has an odd-graceful labeling and every graph with an odd cycle is not odd-graceful. She also proved the following graphs are odd-graceful: P_{n}; C_{n} if and only if n is even; K_{m,n}; combs P_{n} ⊙ K_{1} (graphs obtained by joining a single pendant edge to each vertex of P_{n}); books; crowns C_{n} ⊙ K_{1} (graphs obtained by joining a single pendant edge to each vertex of C_{n}) if and only if n is even; the disjoint union of copies of C_{4}; the one-point union of copies of C_{4}; C_{n} × K_{2} if and only if n is even; caterpillars; rooted trees of height 2; the graphs obtained from P_{n} (n ≥ 3) by adding exactly two leaves at each vertex of degree 2 of P_{n}; the graphs obtained from P_{n} × P_{2} by deleting an edge that joins to end points of the P_{n} paths; the graphs obtained from a star by adjoining to each end vertex the path P_{3} or by adjoining to each end vertex the path P_{4}. She conjectures that all trees are odd-graceful and proves the conjecture for all trees with order up to 10. Barrientos [139] has extended this to trees of order up to 12. Eldergill [295] generalized Gnanajothi’s result on stars by showing that the graphs obtained by joining one end point from each of any odd number of paths of equal length is odd-graceful. He also proved that the onepoint union of any number of copies of C_{6} is odd-graceful. Kathiresan [478] has shown that ladders and graphs obtained from them by subdividing each step exactly once are odd-graceful. Barrientos [142] and [139] has proved the following graphs are odd-graceful: every forest whose components are caterpillars; every tree with diameter at most five is odd-graceful; and all disjoint unions of caterpillars. He conjectures that every bipartite graph is odd-graceful. Seoud, Diab, and Elsakhawi [761] have shown that a connected complete r-partite graph is odd-graceful if and only if r = 2 and that the join of any two connected graphs is not odd-graceful.

Sekar [753] has shown the following graphs are odd-graceful: C_{m} ⊙ P_{n} (the graph obtained by identifying an end point of P_{n} with every vertex of C_{m}) where n ≥ 3 and m is even; P_{a,b} when a ≥ 2 and b is odd (see §2.7); P_{2,b} and b ≥ 2; P_{4,b} and b ≥ 2; P_{a,b} when a and b are even and a ≥ 4 and b ≥ 4; P_{4r+1,4r+2}; P_{4r−1,4r}; all n-polygonal snakes with n even; C_{n}^{(t)} (see §2.2 for the definition); graphs obtained by beginning with C_{6} and repeatedly forming the one-point union with additional copies of C_{6} in succession; graphs obtained by beginning with C_{8} and repeatedly forming the one-point union with additional copies of C_{8} in succession; graphs obtained from even cycles by identifying a vertex of the cycle with the endpoint of a star; C_{6,n} and C_{8,n} (see §2.7); the splitting graph of P_{n} (see §2.7) the splitting graph of C_{n}, n even; lobsters, banana trees, and regular bamboo trees (see §2.1).

Yao, Cheng, Zhongfu, and Yao [989] have shown the following: if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and T_{2} is a caterpillar, then T is odd-graceful; and if a tree T has a vertex of degree at least |T |/2, then T is odd-graceful. They conjecture that for trees, the properties of being Skolem-graceful and odd-graceful are equivalent.

Gao [354] has proved the following graphs are odd-graceful: the union of any number of paths; the union of any number of stars; the union of any number of stars and paths; C_{m} ∪P_{n}; C_{m} ∪C_{n}; and the union of any number of cycles each of which has order divisible by 4.

In [238] Chawathe and Krishna extend the definition of odd-gracefulness to countably infinite graphs and show that all countably infinite bipartite graphs that are connected and locally finite have odd-graceful labelings.

3.6 Graceful-like Labelings

As a means of attacking graph decomposition problems, Rosa [733] invented another analogue of graceful labelings by permitting the vertices of a graph with q edges to assume labels from the set {0, 1, …, q +1}, while the edge labels induced by the absolute value of the difference of the vertex labels are {1, 2, …, q − 1, q} or {1, 2, …, q − 1, q + 1}. He calls these ρ̂-labelings. Frucht [336] used the term nearly graceful labeling instead of ρ̂-labelings. Frucht [336] has shown that the following graphs have nearly graceful labelings with edge labels from {1, 2, …, q − 1, q + 1}: P_{m} ∪ P_{n}; S_{m} ∪ S_{n}; S_{m} ∪ P_{n}; G ∪ K_{2} where G is graceful; and C_{3} ∪ K_{2} ∪ S_{m} where m is even or m ≡ 3 (mod 14). Seoud and Elsakhawi [762] have shown that all cycles are nearly graceful. Barrientos [133] proved that C_{n} is nearly graceful with edge labels 1, 2, …, n − 1, n + 1 if and only if n ≡ 1 or 2 (mod 4). Rosa [735] conjectured that triangular snakes with t ≡ 0 or 1 (mod 4) blocks are graceful and those with t ≡ 2 or 3 (mod 4) blocks are nearly graceful (a parity condition ensures that the graphs in the latter case cannot be graceful). Moulton [669] proved Rosa’s conjecture while introducing the slightly stronger concept of almost graceful by permitting the vertex labels to come from {0, 1, 2, …, q − 1, q + 1} while the edge labels are 1, 2, …, q − 1, q, or 1, 2, …, q − 1, q + 1. Seoud and Elsakhawi [762] and [763] have shown that the following graphs are almost graceful: C_{n}; P_{n} + \overline{K_{m}}; P_{n} + K_{1,m}; K_{m,n}; K_{1,m,n}; K_{2,2,m}; K_{1,1,m,n}; P_{n} × P_{3} (n ≥ 3); K_{5} ∪ K_{1,n}; K_{6} ∪ K_{1,n}, and ladders. The symmetric product G_{1} ⊕ G_{2} of G_{1} and G_{2} is the graph with vertex set V (G_{1}) × V (G_{2}) and edge set {(u_{1}, v_{1})(u_{2}, v_{2})} where u_{1}u_{2} is an edge in G_{1} or v_{1}v_{2} is an edge in G_{2} but not both u_{1}u_{2} is an edge in G_{1} and v_{1}v_{2} is an edge in G_{2}. In [763] Seoud and Elsakhawi show that P_{2} ⊕ \overline{K_{2}} (n ≥ 2) is arbitrarily graceful.

For a graph G with p vertices, q edges, and 1 ≤ k ≤ q, Eshghi [310] defines a holey αlabeling with respect to k as an injective vertex labeling f for which f(v) ∈ {1, 2, …, q+1} for all v, {|f(u) − f(v)| | for all edges uv} = {1, 2, …, k − 1, k + 1, …, q + 1}, and there exist an integer γ with 0 ≤ γ ≤ q such that min{f(u), f(v)} ≤ γ ≤ max{f(u), f(v)}. He proves the following: P_{n} has a holey α-labeling with respect to all k; C_{n} has a holey αlabeling with respect to k if and only if either n ≡ 2 (mod 4), k is even, and (n, k) ≠ (10, 6), or n ≡ 0 (mod 4) and k is odd.

Recall from Section 2.2 that a kC_{n}-snake is a connected graph with k blocks whose block-cutpoint graph is a path and each of the k blocks is isomorphic to C_{n}. In addition to his results on the graceful kC_{n}-snakes given in Section 2.2, Barrientos [136] proved that when k is odd the linear kC_{6}-snake is nearly graceful and that C_{m} ∪K_{1,n} is nearly graceful when m = 3, 4, 5, and 6.

Yet another kind of labeling introduced by Rosa in his 1967 paper [733] is a ρ-labeling. A ρ-labeling (or ρ-valuation) of a graph is an injection from the vertices of the graph with q edges to the set {0, 1, …, 2q}, where if the edge labels induced by the absolute value of the difference of the vertex labels are a_{1}, a_{2}, …, a_{q}, then a_{i} = i or a_{i} = 2q + 1 − i. Rosa [733] proved that a cyclic decomposition of the edge set of the complete graph K_{2q+1} into subgraphs isomorphic to a given graph G with q edges exists if and only if G has a ρ-labeling. (A decomposition of K_{n} into copies of G is called cyclic if the automorphism group of the decomposition itself contains the cyclic group of order n.) It is known that every graph with at most 11 edges has a ρ-labeling and that all lobsters have a ρ-labeling (see [225]). Donovan, El-Zanati, Vanden Eyden, and Sutinuntopas [284] prove that rC_{m} has a ρ-labeling (or a more restrictive labeling) when r ≤ 4. They conjecture that every 2-regular graph has a ρ-labeling. Aguado, El-Zanati, Hake, Stob, and Yayla [36] give a ρ-labeling of C_{r} ∪ C_{s} ∪ C_{t} for each of the cases where r ≡ 0, s ≡ 1, t ≡ 1 (mod 4); r ≡ 0, s ≡ 3, t ≡ 3 (mod 4); and r ≡ 1, s ≡ 1, t ≡ 3 (mod 4); (iv) r ≡ 1, s ≡ 2, t ≡ 3 (mod 4); (v) r ≡ 3, s ≡ 3, t ≡ 3 (mod 4). Caro, Roditty, and Schőnheim [225] provide a construction for the adjacency matrix for every graph that has a ρ-labeling. They ask the following question: If H is a connected graph having a ρ-labeling and q edges and G is a new graph with q edges constructed by breaking H up into disconnected parts, does G also have a ρ-labeling? Kézdy [485] defines a stunted tree as one whose edges can be labeled with e_{1}, e_{2}, …, e_{n} so that e_{1} and e_{2} are incident and, for all j = 3, 4, …, n, edge e_{j} is incident to at least one edge e_{k} satisfying 2k ≤ j − 1. He uses Alon’s “Combinatorial Nullstellensatz” to prove that if 2n + 1 is prime, then every stunted tree with n edges has a ρ-labeling.

In their investigation of cyclic decompositions of complete graphs El-Zanati, Vanden Eynden, and Punnim [304] introduced two kinds of labelings. They say a bipartite graph G with n edges and partite sets A and B has a θ-labeling h if h is a one-to-one function from V (G) to {0, 1, …, 2n} such that {|h(b) − h(a)| ab ∈ E(G), a ∈ A, b ∈ B} = {1, 2, …, n}. They call h a ρ^{+}-labeling of G if h is a one-to-one function from V (G) to {0, 1, …, 2n} and the integers h(x) − h(y) are distinct modulo 2n + 1 taken over all ordered pairs (x, y) where xy is an edge in G, and h(b) > h(a) whenever a ∈ A, b ∈ B and ab is an edge in G. Note that θ-labelings are ρ^{+}-labelings and ρ^{+}-labelings are ρ-labelings. They prove that if G is a bipartite graph with n edges and a ρ^{+}-labeling, then for every positive integer x there is a cyclic G-decomposition of K_{2nx+1}. They prove the following graphs have ρ^{+}-labelings: trees of diameter at most 5, C_{2n}, lobsters, and comets (that is, graphs obtained from stars by replacing each edge by a path of some fixed length). They also prove that the disjoint union of graphs with α-labelings have a θ-labeling and conjecture that all forests have ρ-labelings.

A σ-labeling of G(V, E) is a one-to-one function f from V to {0, 1, …, 2|E|} such that {|f(u) − f(v)| | uv ∈ E(G)} = {1, 2, …, |E|}. Such a labeling of G yields cyclic Gdecompositions of K_{2n+1} and of K_{2n+2} − F, where F is a 1-factor of K_{2n+2}. El-Zanati and Vanden Eynden (see [35]) have conjectured that that every 2-regular graph with n edges has a ρ-labeling and, if n ≡ 0 or 3 (mod 4), then every 2-regular graph has a σ-labeling. Aguado and El-Zanati [35] have proved that the latter conjecture holds when the graph has at most three components.

Given a bipartite graph G with partite sets X and Y and graphs H_{1} with p vertices and H_{2} with q vertices, Fronček and Winters [333] define the bicomposition of G and H_{1} and H_{2}, G[H_{1}, H_{2}], as the graph obtained from G by replacing each vertex of X by a copy of H_{1}, each vertex of Y by a copy of H_{2}, and every edge xy by a graph isomorphic to K_{p,q} with the partite sets corresponding to the vertices x and y. They prove that if G is a bipartite graph with n edges and G has a θ-labeling that maps the vertex set V = X ∪ Y into a subset of \{0, 1, 2, …, 2n\}, then the bicomposition G[\overline{K_{p}}, \overline{K_{q}}] has a θ-labeling for every p, q ≥ 1. As corollaries they have: if a bipartite graph G with n edges and at most n + 1 vertices has a gracious labeling (see §3.1), then the bicomposition graph G[\overline{K_{p}}, \overline{K_{q}}] has a gracious labeling for every p, q ≥ 1, and if a bipartite graph G with n edges has a θ-labeling, then for every p, q ≥ 1, the bicomposition G[\overline{K_{p}}, \overline{K_{q}}] decomposes the complete graph K_{2npq+1}.

Blinco, El-Zanati, and Vanden Eynden [174] call a non-bipartite graph almost-bipartite if the removal of some edge results in a bipartite graph. For these kinds of graphs G they call a labeling f a γ-labeling of G if the following conditions are met: f is a ρ-labeling; G is tripartite with vertex tripartition A, B, C with C = \{c\} and \overline{b} ∈ B such that \{\overline{b}, c\} is the unique edge joining an element of B to c; if {a, v} is an edge of G with a ∈ A, then f(a) < f(v); and f(c)−f(\overline{b}) = n. (In § 3.3 the term γ-labeling is used for a different kind of labeling.) They prove that if an almost-bipartite graph G with n edges has a γ-labeling then there is a cyclic G-decomposition of K_{2nx+1} for all x. They prove that all odd cycles with more than 3 vertices have a γ-labeling and that C_{3} ∪ C_{4m} has a γ-labeling if and only if m > 1. In [205] Bunge, El-Zanati, and Vanden Eynden prove that every 2-regular almost bipartite graph other than C_{3} and C_{3} ∪ C_{4} have a γ-labeling.

In [174] Blinco, El-Zanati, and Vanden Eynden consider a slightly restricted ρ^{+}- labeling for a bipartite graph with partite sets A and B by requiring that there exists a number λ with the property that ρ^{+}(a) ≤ λ for all a ∈ A and ρ^{+}(b) > λ for all b ∈ B. They denote such a labeling by ρ^{++}. They use this kind of labeling to show that if G is a 2-regular graph of order n in which each component has even order then there is a cyclic G-decomposition of K_{2nx+1} for all x. They also conjecture that every bipartite graph has a ρ-labeling and every 2-regular graph has a ρ-labeling.

Dufour [294] and Eldergill [295] have some results on the decomposition of complete graphs using labeling methods. Balakrishnan and Sampathkumar [123] showed that for each positive integer n the graph \overline{K_{n}} + 2K_{2} admits a ρ-labeling. Balakrishnan [119] asks if it is true that \overline{K_{n}} +mK_{2} admits a ρ-labeling for all n and m. Fronček [330] and Fronček and Kubesa [332] have introduced several kinds of labelings for the purpose of proving the existence of special kinds of decompositions of complete graphs into spanning trees.

For (p, q)-graphs with p = q + 1, Frucht [336] has introduced a stronger version of almost graceful graphs by permitting as vertex labels {0, 1, …, q − 1, q + 1} and as edge labels {1, 2, …, q}. He calls such a labeling pseudograceful. Frucht proved that P_{n} (n ≥ 3), combs, sparklers (i.e., graphs obtained by joining an end vertex of a path to the center of a star), C_{3} ∪ P_{n} (n ≠ 3), and C_{4} ∪ P_{n} (n ≠ 1) are pseudograceful whereas K_{1,n} (n ≥ 3) is not. Kishore [490] proved that C_{s} ∪ P_{n} is pseudograceful when s ≥ 5 and n ≥ (s + 7)/2 and that C_{s} ∪ S_{n} is pseudograceful when s = 3, s = 4, and s ≥ 7. Seoud and Youssef [773] and [769] extended the definition of pseudograceful to all graphs with p ≤ q + 1. They proved that K_{m} is pseudograceful if and only if m = 1, 3, or 4 [769]; K_{m,n} is pseudograceful when n ≥ 2, and P_{m} + \overline{K_{n}} (m ≥ 2) [773] is pseudograceful. They also proved that if G is pseudograceful, then G ∪ K_{m,n} is graceful for m ≥ 2 and n ≥ 2 and G ∪ K_{m,n} is pseudograceful for m ≥ 2, n ≥ 2 and (m, n) ≠ (2, 2) [769]. They ask if G ∪ K_{2,2} is pseudograceful whenever G is. Seoud and Youssef [769] observed that if G is a pseudograceful Eulerian graph with q edges, then q ≡ 0 or 3 (mod 4). Youssef [997] has shown that C_{n} is pseudograceful if and only if n ≡ 0 or 3 (mod 4), and for n > 8 and n ≡ 0 or 3 (mod 4), C_{n} ∪ K_{p,q} is pseudograceful for all p, q ≥ 2 except (p, q) = (2, 2). Youssef [994] has shown that if H is pseudograceful and G has an α-labeling with k being the smaller vertex label of the edge labeled with 1 and if either k + 2 or k − 1 is not a vertex label of G, then G ∪ H is graceful. In [998] Youssef shows that if G is (p, q) pseudograceful graph with p = q + 1, then G ∪ S_{m} is Skolem-graceful. As a corollary he obtains that for all n ≥ 2, P_{n} ∪ S_{m} is Skolem-graceful if and only if n ≥ 3 or n = 2 and m is even.

McTavish [647] has investigated labelings of graphs with q edges where the vertex and edge labels are from {0, …, q, q + 1}. She calls these ρ̃-labelings. Graphs that have ρ̃-labelings include cycles and the disjoint union of P_{n} or S_{n} with any graceful graph.

Frucht [336] has made an observation about graceful labelings that yields nearly graceful analogs of α-labelings and weakly α-labelings in a natural way. Suppose G(V, E) is a graceful graph with the vertex labeling f. For each edge xy in E, let [f(x), f(y)] (where f(x) ≤ f(y)) denote the interval of real numbers r with f(x) ≤ r ≤ f(y). Then the intersection ∩[f(x), f(y)] over all edges xy ∈ E is a unit interval, a single point, or empty. Indeed, if f is an α-labeling of G then the intersection is a unit interval; if f is a weakly α-labeling, but not an α-labeling, then the intersection is a point; and, if f is a graceful but not a weakly α-labeling, then the intersection is empty. For nearly graceful labelings, the intersection also gives three distinct classes.

Singh and Devaraj [839] call a graph G with p vertices and q edges triangular graceful if there is an injection f from V (G) to {0, 1, 2, …, T_{q}} where T_{q} is the qth triangular number and the labels induced on each edge uv by |f(u)−f(v)| are the first q triangular numbers. They prove the following graphs are triangular graceful: paths, level 2 rooted trees, olive trees (see § 2.1 for the definition), complete n-ary trees, double stars, caterpillars, C_{4n}, C_{4n} with pendent edges, the one-point union of C_{3} and P_{n}, and unicyclic graphs that have C_{3} as the unique cycle. They prove that wheels, helms, flowers (see §2.2 for the definition) and K_{n} with n ≥ 3 are not triangular graceful. They conjecture that all trees are triangular graceful.

Van Bussel [921] considered two kinds of relaxations of graceful labelings as applied to trees. He called a labeling range-relaxed graceful it is meets the same conditions as a graceful labeling except the range of possible vertex labels and edge labels are not restricted to the number of edges of the graph (the edges are distinctly labeled but not necessarily labeled 1 to q where q is the number of edges). Similarly, he calls a labeling vertex-relaxed graceful if it satisfies the conditions of a graceful labeling while permitting repeated vertex labels. He proves that every tree T with q edges has a range-relaxed graceful labeling with the vertex labels in the range 0, 1, …, 2q−d where d is the diameter of T and that every tree on n vertices has a vertex-relaxed graceful labeling such that the number of distinct vertex labels is strictly greater than n/2.

Sekar [753] calls an injective function ϕ from the vertices of a graph with q edges to {0, 1, 3, 4, 6, 7, …, 3(q − 1), 3q − 2} one modulo three graceful if the edge labels induced by labeling each edge uv with |ϕ(u) − ϕ(v)| is {1, 4, 7, …, 3q − 2}. He proves that the following graphs are one modulo three graceful: P_{m}; C_{n} if and only if n ≡ 0 mod 4; K_{m,n}; C_{2n}^{(2)} (the one-point union of two copies of C_{2n}); C_{n}^{(t)} for n = 4 or 8 and t > 2; C_{6}^{(t)} and t ≥ 4; caterpillars, stars, lobsters; banana trees, rooted trees of height 2; ladders; the graphs obtained by identifying the endpoints of any number of copies of P_{n}; the graph obtained by attaching pendent edges to each endpoint of two identical stars and then identifying one endpoint from each of these graphs; the graph obtained by identifying a vertex of C_{4k+2} with an endpoint of a star; n-polygonal snakes (see §2.2) for n ≡ 0 (mod 4); n-polygonal snakes for n ≡ 2 (mod 4) where the number of polygons is even; crowns C_{n} ⊙ K_{1} for n even; C_{2n} ⊙ P_{m} (C_{2n} with P_{m} attached at each vertex of the cycle) for m ≥ 3; chains of cycles (see §2.2) of the form C_{4,m}, C_{6,2m}, and C_{8,m}. He conjectures that every one modulo three graceful graph is graceful.

In [193] Brešar and Klavžar define a natural extension of graceful labelings of certain tree subgraphs of hypercubes. A subgraph H of a graph G is called isometric if for every two vertices u, v of H, there exists a shortest u-v path that lies in H. The isometric subgraphs of hypercubes are called partial cubes. Two edges xy, uv of G are in Θ-relation if

d_{G}(x, u) + d_{G}(y, v) ≠ d_{G}(x, v) + d_{G}(y, u).

A Θ-relation is an equivalence relation that partitions E(G) into Θ-classes. A Θ-graceful labeling of a partial cube G on n vertices is a bijection f: V (G) → {0, 1, …, n − 1} such that, under the induced edge labeling, all edges in each Θ-class of G have the same label and distinct Θ-classes get distinct labels. They prove that several classes of partial cubes are Θ-graceful and the Cartesian product of Θ-graceful partial cubes is Θ-graceful. They also show that if there exists a class of partial cubes that contains all trees and every member of the class admits a Θ-graceful labeling then all trees are graceful.

3.7 Cordial Labelings

Cahit [211] has introduced a variation of both graceful and harmonious labelings. Let f be a function from the vertices of G to {0, 1} and for each edge xy assign the label |f(x) − f(y)|. Call f a cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1, and the number of edges labeled 0 and the number of edges labeled 1 differ at most by 1. Cahit [212] proved the following: every tree is cordial; K_{n} is cordial if and only if n ≤ 3; K_{m,n} is cordial for all m and n; the friendship graph C_{3}^{(t)} (i.e., the one-point union of t 3-cycles) is cordial if and only if t ≢ 2 (mod 4); all fans are cordial; the wheel W_{n} is cordial if and only if n ≢ 3 (mod 4) (see also [291]); maximal outerplanar graphs are cordial; and an Eulerian graph is not cordial if its size is congruent to 2 (mod 4). Kuo, Chang, and Kwong [521] determine all m and n for which mK_{n} is cordial. Youssef [998] proved that every Skolem-graceful graph (see §3.4 for the definition) is cordial. Liu and Zhu [621] proved that a 3-regular graph of order n is cordial if and only if n ≢ 4 (mod 8).

A k-angular cactus is a connected graph all of whose blocks are cycles with k vertices. In [212] Cahit proved that a k-angular cactus with t cycles is cordial if and only if kt ≢ 2 (mod 4). This was improved by Kirchherr [488] who showed any cactus whose blocks are cycles is cordial if and only if the size of the graph is not congruent to 2 (mod 4). Kirchherr [489] also gave a characterization of cordial graphs in terms of their adjacency matrices. Ho, Lee, and Shee [439] proved: P_{n} × C_{4m} is cordial for all m and all odd n; the composition G and H is cordial if G is cordial and H is cordial and has odd order and even size (see §2.3 for definition of composition); for n ≥ 4 the composition C_{n}[K_{2}] is cordial if and only if n ≢ 2 (mod 4); the Cartesian product of two cordial graphs of even size is cordial. He, Lee, and Shee [438] showed that a unicyclic graph is cordial unless it is C_{4k+2} and that the generalized Petersen graph (see §2.7 for the definition) P (n, k) is cordial if and only if n ≢ 2 (mod 4). Du [291] determines the maximal number of edges in a cordial graph of order n and gives a necessary condition for a k-regular graph to be cordial. Riskin [726] proved that Möbius ladders M_{n} (see §2.3 for the definition) are cordial if and only if n ≥ 3 and n ≢ 2 (mod 4). (See also [763].)

Seoud and Abdel Maqusoud [759] proved that if G is a graph with n vertices and m edges and every vertex has odd degree, then G is not cordial when m + n ≡ 2 (mod 4). They also prove the following: for m ≥ 2, C_{n} × P_{m} is cordial except for the case C_{4k+2} ×P_{2}; P_{n}^{2} is cordial for all n; P_{n}^{3} is cordial if and only if n ≠ 4; and P_{n}^{4} is cordial if and only if n ≠ 4, 5, or 6. Seoud, Diab, and Elsakhawi [761] have proved the following graphs are cordial: P_{n} +P_{m} for all m and n except (m, n) = (2, 2); C_{m} +C_{n} if m ≢ 0 (mod 4) and n ≠ 2 (mod 4); C_{n} + K_{1,m} for n ≢ 3 (mod 4) and odd m except (n, m) = (3, 1); C_{n} + \overline{K_{m}} when n is odd, and when n is even and m is odd; K_{1,m,n}; K_{2,2,m}; the n-cube; books B_{n} if and only if n ≢ 3 (mod 4); B(3, 2, m) for all m; B(4, 3, m) if and only if m is even; and B(5, 3, m) if and only if m ≢ 1 (mod 4) (see §2.4 for the notation B(n, r, m)).

Diab [282] proved the following graphs are cordial: C_{m} + P_{n} if and only if (m, n) ≠ (3, 3), (3, 2), or (3,1); P_{m} + K_{1,n} if and only if (m, n) ≠ (1, 2); P_{m} ∪ K_{1,n} if and only if (m, n) ≠ (1, 2); C_{m} ∪K_{1,n}; C_{m} +\overline{K_{n}} for all m and n except m ≡ 3 (mod 4) and n odd, and m ≡ 2 (mod 4) and n even; C_{m} ∪ \overline{K_{n}} for all m and n except m ≡ 2 (mod 4); P_{m} + \overline{K_{n}}; and P_{m} ∪ \overline{K_{n}}.

Youssef [999] has proved the following: If G and H are cordial and one has even size, then G ∪ H is cordial; if G and H are cordial and both have even size, then G + H is cordial; if G and H are cordial and one has even size and either one has even order, then G + H is cordial; C_{m} ∪ C_{n} is cordial if and only if m + n ≢ 2 (mod 4); mC_{n} is cordial if and only if mn ≢ 2 (mod 4); C_{m} + C_{n} is cordial if and only if (m, n) ≠ (3, 3) and \{m (mod 4), n (mod 4)\} ≠ \{0, 2\}; and if P_{n}^{k} is cordial, then n ≥ k + 1 + \sqrt{k − 2}. He conjectures that this latter condition is also sufficient. He confirms the conjecture for k = 5, 6, 7, 8, and 9.

Lee and Liu [544] have shown that the complete n-partite graph is cordial if and only if at most three of its partite sets have odd cardinality (see also [291]). Lee, Lee, and Chang [529] prove the following graphs are cordial: the Cartesian product of an arbitrary number of paths; the Cartesian product of two cycles if and only if at least one of them is even; and the Cartesian product of an arbitrary number of cycles if at least one of them has length a multiple of 4 or at least two of them are even.

Shee and Ho [804] have investigated the cordiality of the one-point union of n copies of various graphs. For C_{m}^{(n)}, the one-point union of n copies of C_{m}, they prove:

(i) If m ≡ 0 (mod 4), then C_{m}^{(n)} is cordial for all n;

(ii) If m ≡ 1 or 3 (mod 4), then C_{m}^{(n)} is cordial if and only if n ≢ 2 (mod 4); (iii) If m ≡ 2 (mod 4), then C_{m}^{(n)} is cordial if and only if n is even.

For K_{m}^{(n)}, the one-point union of n copies of K_{m}, Shee and Ho [804] prove:

(i) If m ≡ 0 (mod 8), then K_{m}^{(n)} is not cordial for n ≡ 3 (mod 4); (ii) If m ≡ 4 (mod 8), then K_{m}^{(n)} is not cordial for n ≡ 1 (mod 4); (iii) If m ≡ 5 (mod 8), then K_{m}^{(n)} is not cordial for all odd n; (iv) K_{4}^{(n)} is cordial if and only if n ≢ 1 (mod 4);

(v) K_{5}^{(n)} is cordial if and only if n is even;

(vi) K_{6}^{(n)} is cordial if and only if n > 2;

(vii) K_{7}^{(n)} is cordial if and only if n ≢ 2 (mod 4);

(viii) K_{n}^{(2)} is cordial if and only if n has the form p^{2} or p^{2} + 1.

In his 2001 Ph. D. thesis Selvaraju [754] proves that the one-point union of any number of copies of a complete bipartite graph is cordial. Benson and Lee [154] have investigated the regular windmill graphs K_{m}^{(n)} and determined precisely which ones are cordial for m < 14.

For W_{m}^{(n)}, the one-point union of n copies of the wheel W_{m} with the common vertex being the center, Shee and Ho [804] show:

(i) If m ≡ 0 or 2 (mod 4), then W_{m}^{(n)} is cordial for all n;

(ii) If m ≡ 3 (mod 4), then W_{m}^{(n)} is cordial if n ≢ 1 (mod 4);

(iii) If m ≡ 1 (mod 4), then W_{m}^{(n)} is cordial if n ≢ 3 (mod 4).

For all n and all m > 1 Shee and Ho [804] prove F_{m}^{(n)}, the one-point union of n copies of the fan F_{m} = P_{m} + K_{1} with the common point of the fans being the center, is cordial (see also [601]). The flag F l_{m} is obtained by joining one vertex of C_{m} to an extra vertex called the root. Shee and Ho [804] show all F l_{m}^{(n)}, the one-point union of n copies of F l_{m} with the common point being the root, are cordial.

Andar, Boxwala, and Limaye [44], [45], and [48] have proved the following graphs are cordial: helms; closed helms; generalized helms obtained by taking a web (see 2.2 for the definitions) and attaching pendent vertices to all the vertices of the outermost cycle in the case that the number cycles is even; flowers, which are obtained by joining the vertices of degree one of a helm to the central vertex; sunflower graphs, which are obtained by taking a wheel with the central vertex v_{0} and the n-cycle v_{1}, v_{2}, …, v_{n} and additional vertices w_{1}, w_{2}, …, w_{n} where w_{i} is joined by edges to v_{i}, v_{i+1}, where i + 1 is taken modulo n: multiple shells (see §2.2); and one point unions of helms, closed helms, flowers, gears, and sunflower graphs, where in each case the central vertex is the common vertex.

Du [292] proved that the disjoint union of n ≥ 2 wheels is cordial if and only if n is even or n is odd and the number of vertices of in each cycle is not 0 (mod 4). or n is odd and the number of vertices of in each cycle is not 3 (mod 4).

Elumalai and Sethurman [298] proved: cycles with parallel cords are cordial and ncycles with parallel P_{k}-chords (see §2.2 for the definition) are cordial for any odd positive integer k at least 3 and any n ≢ 2 (mod 4) of length at least 4. They call a graph H a even-multiple subdivision graph of a graph G if it is obtained from G by replacing every edge uv of G by a pair of paths of even length starting at u and ending at v. They prove that every even-multiple subdivision graph is cordial and that every graph is a subgraph of a cordial graph. In [950] Wen proves that generalized wheels C_{n} + mK_{1} are cordial when m is even and n ≢ 2 (mod 4) and when m is odd and n ≢ 3 (mod 4).

In [48] Andar et al. define a t-ply graph P_{t}(u, v) as a graph consisting of t internally disjoint paths joining vertices u and v. They prove that P_{t}(u, v) is cordial except when it is Eulerian and the number of edges is congruent to 2 (mod 4). In [49] Andar, Boxwala, and Limaye prove that the one-point union of any number of plys with an endpoint as the common vertex is cordial if and only if it is not Eulerian and the number of edges is congruent to 2 (mod 4). They further prove that the path union of shells obtained by joining any point of one shell to any point of the next shell is cordial; graphs obtained by attaching a pendant edge to the common vertex of the cords of a shell are cordial; and cycles with one pendant edge are cordial.

For a graph G and a positive integer t, Andar, Boxwala, and Limaye [46] define the t-uniform homeomorph P_{t}(G) of G as the graph obtained from G by replacing every edge of G by vertex disjoint paths of length t. They prove that if G is cordial and t is odd, then P_{t}(G) is cordial; if t ≡ 2 (mod 4) a cordial labeling of G can be extended to a cordial labeling of P_{t}(G) if and only if the number of edges labeled 0 in G is even; and when t ≡ 0 (mod 4) a cordial labeling of G can be extended to a cordial labeling of P_{t}(G) if and only if the number of edges labeled 1 in G is even. In [47] Ander et al. prove that P_{t}(K_{2n}) is cordial for all t ≥ 2 and that P_{t}(K_{2n+1}) is cordial if and only if t ≡ 0 (mod 4) or t is odd and n ≢ 2 (mod 4), or t ≡ 2 (mod 4) and n is even.

In [49] Andar, Boxwala, and Limaya show that a cordial labeling of G can be extended to a cordial labeling of the graph obtained from G by attaching 2m pendant edges at each vertex of G. For a binary labeling g of a graph G let v_{g}(j) denote the number of vertices labeled with j and e_{g}(j) denote the number edges labeled with j. Let i(G) = min{|e_{g}(0) − e_{g}(1)|} taken over all binary labelings g of G with |v_{g}(0) − v_{g}(1)| ≤ 1. Andar et al. also prove that a cordial labeling g of a graph G with p vertices can be extended to a cordial labeling of the graph obtained from G by attaching 2m + 1 pendant edges at each vertex of G if and only if G does not satisfy either of the conditions: (1) G has an even number of edges and p ≡ 2 (mod 4); (2) G has an odd number of edges and either p ≡ 1 (mod 4) with e_{g}(1) = e_{g}(0) + i(G) or n ≡ 3 (mod 4) and e_{g}(0) = e_{g}(1) + i(G). Andar, Boxwala, and Limaye [50] also prove: if g is a binary labeling of a graph G then g can be extended to a cordial labeling of G ⊙ \overline{K_{2m}} if and only if n is odd and i(G) ≡ 2 (mod 4); K_{n} ⊙ \overline{K_{2m}} is cordial if and only if n ≠ 4 (mod 8); K_{n} ⊙ \overline{K_{2m+1}} is cordial if and only if n ≠ 7 (mod 8); if g is a binary labeling on a graph G with n vertices, then g can be extended to a cordial labeling of G ⊙ C_{t} if t ≠ 3 mod 4, n is odd and e_{g}(0) = e_{g}(1). For any binary labeling g of a graph G they also characterize in terms of of i(G) when g can be extended to graphs of the form G ⊙ \overline{K_{2m+1}}.

For graphs G_{1}, G_{2}, …, G_{n} (n ≥ 2) that are all copies of a fixed graph G, Shee and Ho [805] call a graph obtained by adding an edge from G_{i} to G_{i+1} for i = 1, …, n − 1 a pathunion of G (the resulting graph may depend on how the edges are chosen). Among their results they show the following graphs are cordial: path-unions of cycles; path-unions of n copies of K_{m} when m = 4, 6, or 7; path-unions of three or more copies of K_{5}; and path-unions of two copies of K_{m} if and only if m−2, m, or m+2 is a perfect square. They also show that there exist cordial path-unions of wheels, fans, unicyclic graphs, Petersen graphs, trees, and various compositions.

Lee and Liu [544] give the following general construction for the forming of cordial graphs from smaller cordial graphs. Let H be a graph with an even number of edges and a cordial labeling such that the vertices of H can be divided into t parts H_{1}, H_{2}, …, H_{t} each consisting of an equal number of vertices labeled 0 and vertices labeled 1. Let G be any graph and G_{1}, G_{2}, …, G_{t} be any t subsets of the vertices of G. Let (G, H) be the graph that is the disjoint union of G and H augmented by edges joining every vertex in G_{i} to every vertex in H_{i} for all i. Then G is cordial if and only if (G, H) is. From this it follows that: all generalized fans F_{m,n} = \overline{K_{m}} + P_{n} are cordial; the generalized bundle B_{m,n} is cordial if and only if m is even or n ≢ 2 (mod 4) (B_{m,n} consists of 2n vertices v_{1}, v_{2}, …, v_{n}, u_{1}, u_{2}, …, u_{n} with an edge from v_{i} to u_{i} and 2m vertices x_{1}, x_{2}, … x_{m}, y_{1}, y_{2}, …, y_{m} with x_{i} joined to v_{i} and y_{i} joined to u_{i}); if m is odd the generalized wheel W_{m,n} = \overline{K_{m}} + C_{n} is cordial if and only if n ≢ 3 (mod 4). If m is even, W_{m,n} is cordial if and only if n ≢ 2 (mod 4); a complete k-partite graph is cordial if and only if the number of parts with an odd number of vertices is at most 3.

Sethuraman and Selvaraju [799] have shown that certain cases of the union of any number of copies of K_{4} with one or more edges deleted and one edge in common are cordial. Youssef [1000] has shown that the kth power of C_{n} is cordial for all n when k ≡ 2 (mod 4) and for all even n when k ≡ 0 (mod 4).

Cahit [217] calls a graph H-cordial if it is possible to label the edges with the numbers from the set {1, −1} in such a way that, for some k, at each vertex v the sum of the labels on the edges incident with v is either k or −k and the inequalities |v(k) − v(−k)| ≤ 1 and |e(1) − e(−1)| ≤ 1 are also satisfied, where v(i) and e(j) are, respectively, the number of vertices labeled with i and the number of edges labeled with j. He calls a graph H_{n}-cordial if it is possible to label the edges with the numbers from the set {±1, ±2, …, ±n} in such a way that, at each vertex v the sum of the labels on the edges incident with v is in the set {±1, ±2, …, ±n} and the inequalities |v(i) − v(−i)| ≤ 1 and |e(i) − e(−i)| ≤ 1 are also satisfied for each i with 1 ≤ i ≤ n. Among Cahit’s results are: K_{n,n} is H-cordial if and only if n > 2 and n is even; and K_{m,n}, m ≠ n, is H-cordial if and only if n ≡ 0 (mod 4), m is even and m > 2, n > 2. Unfortunately, Ghebleh and Khoeilar [365] have shown that other statements in Cahit’s paper are incorrect. In particular, Cahit states that K_{n} is H-cordial if and only if n ≡ 0 (mod 4); W_{n} is H-cordial if and only if n ≡ 1 (mod 4); and K_{n} is H_{2}-cordial if and only if n ≡ 0 (mod 4) whereas Ghebleh and Khoeilar instead prove that K_{n} is H-cordial if and only if n ≡ 0 or 3 (mod 4) and n ≠ 3; W_{n} is H-cordial if and only if n is odd; K_{n} is H_{2}-cordial if n ≡ 0 or 3 (mod 4); and K_{n} is not H_{2}-cordial if n ≡ 1 (mod 4). Ghebleh and Khoeilar also prove every wheel has an H_{2}-cordial labeling. Cahit generalizes the notion of H-cordial labelings in [217].

Cahit and Yilmaz [221] call a graph E_{k}-cordial if it is possible to label the edges with the numbers from the set {0, 1, 2, …, k − 1} in such a way that, at each vertex v, the sum modulo k of the labels on the edges incident with v satisfies the inequalities |v(i) − v(j)| ≤ 1 and |e(i) − e(j)| ≤ 1, where v(s) and e(t) are, respectively, the number of vertices labeled with s and the number of edges labeled with t. Cahit and Yilmaz prove the following graphs are E_{3}-cordial: P_{n} (n ≥ 3); stars S_{n} if and only if n ≢ 1 (mod 3); K_{n} (n ≥ 3); C_{n} (n ≥ 3); friendship graphs; and fans F_{n} (n ≥ 3). They also prove that S_{n} (n ≥ 2) is E_{k}-cordial if and only if n ≢ 1 (mod k) when k is odd or n ≢ 1 (mod 2k) when k is even and k ≠ 2.

Bapat and Limaye [131] provide E_{3}-cordial labelings for: K_{n} (n ≥ 3); snakes whose blocks are all isomorphic to K_{n} where n ≡ 0 or 2 (mod 3); the one-point union of any number of copies of K_{n} where n ≡ 0 or 2 (mod 3); graphs obtained by attaching a copy of K_{n} where n ≡ 0 or 3 (mod 3) at each vertex of a path; and K_{m} ⊙ K_{n}. Rani and Sridharan [719] proved: for odd n > 1 and k ≥ 2, P_{n} ⊙ K_{1} is E_{k}-cordial; for n even and n ≠ k/2, P_{n} ⊙ K_{1} is E_{k}-cordial; and certain cases of fans are E_{k}-cordial.

Hovey [441] has introduced a simultaneous generalization of harmonious and cordial labelings. For any Abelian group A (under addition) and graph G(V, E) he defines G to be A-cordial if there is a labeling of V with elements of A such that for all a and b in A when the edge ab is labeled with f(a)+f(b), the number of vertices labeled with a and the number of vertices labeled b differ by at most one and the number of edges labeled with a and the number labeled with b differ by at most one. In the case where A is the cyclic group of order k, the labeling is called k-cordial. With this definition we have: G(V, E) is harmonious if and only if G is |E|-cordial; G is cordial if and only if G is 2-cordial.

Hovey has obtained the following: caterpillars are k-cordial for all k; all trees are k-cordial for k = 3, 4, and 5; odd cycles with pendant edges attached are k-cordial for all k; cycles are k-cordial for all odd k; for k even, C_{2mk+j} is k-cordial when 0 ≤ j ≤ \frac{k}{2} + 2 and when k < j < 2k; C_{(2m+1)k} is not k-cordial; K_{m} is 3-cordial; and, for k even, K_{mk} is k-cordial if and only if m = 1.

Hovey advances the following conjectures: all trees are k-cordial for all k; all connected graphs are 3-cordial; and C_{2mk+j} is k-cordial if and only if j ≠ k, where k and j are even and 0 ≤ j < 2k. The last conjecture was verified by Tao [906]. Tao’s result combined with those of Hovey show that for all positive integers k the n-cycle is k-cordial with the exception that k is even and n = 2mk + k. Tao also proved that the crown with 2mk + j vertices is k-cordial unless j = k is even, and for 4 ≤ n ≤ k the wheel W_{n} is k-cordial unless k ≡ 5 (mod 8) and n = (k + 1)/2.

In [795] Sethuraman and Selvaraju present an algorithm that permits one to start with any non-trivial connected graph G and successively form supersubdivisions (see §2.7 for the definition) that are cordial in the case that every edge in G is replaced by K_{2,m} where m is even. Sethuraman and Selvaraju [794] also show that the one-vertex union of any number of copies of K_{m,n} is cordial and that the one-edge union of k copies of shell graphs C(n, n − 3) (see §2.2) is cordial for all n ≥ 4 and all k. They conjectured that the one-point union of any number of copies of graphs of the form C(n_{i}, n_{i} − 3) for various n_{i} ≥ 4 is cordial. This was proved by Yue, Yuansheng, and Liping in [1005]. Riskin [728] proved that K_{n} is Z_{2} × Z_{2}-cordial if and only if n is at most 3 and K_{m,n} is Z_{2} × Z_{2} cordial if and only if (m, n) ≠ (2, 2).

Cairnie and Edwards [223] have determined the computational complexity of cordial and k-cordial labelings. They prove the conjecture of Kirchherr [489] that deciding whether a graph admits a cordial labeling is NP-complete. As a corollary, this result implies that the same problem for k-cordial labelings is NP-complete. They remark that even the restricted problem of deciding whether connected graphs of diameter 2 have a cordial labeling is also NP-complete.

In [236] Chartrand, Lee, and Zhang introduced the notion of uniformly cordial as follows. Let f be a labeling from V (G) to {0, 1} and for each edge xy define f^{∗}(xy) = |f(x) − f(y)|. For i = 0 and 1, let n_{i}(f) denote the number of vertices v with f(v) = i and m_{i}(f) denote the number of edges e with f^{∗}(e) = i. They call a such a labeling f friendly if |n_{0}(f) − n_{1}(f)| ≤ 1. A graph G for which every friendly labeling is cordial is called uniformly cordial. They prove that a connected graph of order n ≥ 2 is uniformly cordial if and only if n = 3 and G = K_{3}, or n is even and G = K_{1,n−1}.

In [726] Riskin introduced two measures of the noncordiality of a graph. He defines the cordial edge deficiency of a graph G as the minimum number of edges, taken over all friendly labelings of G, needed to be added to G such that the resulting graph is cordial. If a graph G has a vertex labeling f using 0 and 1 such that the edge labeling f_{e} given by f_{e}(xy) = |f(x) − f(y)| has the property that the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1, the cordial vertex deficiency defined as ∞. Riskin proved: the cordial edge deficiency of K_{n} (n > 1) is ⌊\frac{n}{2}⌋ − 1; the cordial vertex deficiency of K_{n} is j − 1 if n = j^{2} + δ, when δ is −2, 0 or 2, and ∞ otherwise. In [726] Riskin determines the cordial edge deficiency and cordial vertex deficiency for the cases when the Möbius ladders and wheels are not cordial. In [727] Riskin determines the cordial edge deficiencies for complete multipartite graphs that are not cordial and obtains a upper bound for their cordial vertex deficiencies.

If f is a binary vertex labeling of a graph G Lee, Liu, and Tan [545] defined a partial edge labeling of the edges of G by f^{∗}(uv) = 0 if f(u) = f(v) = 0 and f^{∗}(uv) = 1 if f(u) = f(v) = 1. They let e_{0}(G) denote the number of edges uv for which f^{∗}(uv) = 0 and e_{1}(G) denote the number of edges uv for which f^{∗}(uv) = 1. They say G is balanced if it has a friendly labeling f such that if |e_{0}(f) − e_{1}(f)| ≤ 1. In the case that the number of vertices labeled 0 and the number of vertices labeled 1 are equal and the number of edges labeled 0 and the number of edges labeled 1 are equal they say the labeling is strongly balanced. They prove: P_{n} is balanced for all n and is strongly balanced if n is even; K_{m,n} is balanced if and only if m and n are even, m and n are odd and differ by at most 2, or exactly one of m or n is even (say n = 2t) and t ≡ −1, 0, 1 (mod |m − n|); and a k-regular graph with p vertices is strongly balanced if and only if p is even and is balanced if and only if p is odd and k = 2. In [502] Kong, Lee, Seah, and Tang show: C_{m} ×P_{n} is balanced if m and n are odd and is strongly balanced if either m or n is even; and C_{m} ⊙ K_{1} is balanced for all m ≥ 3 and strongly balanced if m is even. They also provide necessary and sufficient conditions for a graph to be balanced or strongly balanced. Lee, Lee, and Ng [528] show that stars are balanced if and only if the number of edges of the star is at most 4. Kwong, Lee, Lo, and Wang [524] define a graph G to be uniformly balanced if |e_{0}(f) − e_{1}(f)| ≤ 1 for every vertex lableing f that satisfies if |v_{0}(f) − v_{1}(f)| ≤ 1. They present several ways to construct families of uniformly balanced graphs.

3.8 The Friendly Index–Balance Index

Lee and Ng [549] define the friendly index set of a graph G as FI(G)= {|n_{0}(f) − n_{1}(f)| where f runs over all friendly labelings f of G}. They proved: for any graph G with q edges FI(G) ⊆ {0, 2, 4, …, q} if q is even and FI(G)⊆ {1, 3, …, q} if q is odd; for 1 ≤ m ≤ n, FI(K_{m,n})= {(m − 2i)^{2}| 0 ≤ i ≤ ⌊m/2⌋} if m + n is even; and FI(K_{m,n})= {i(i + 1)| 0 ≤ i ≤ m} if m + n is odd. In [550] Lee and Ng prove the following: FI(C_{2n}) = {0, 4, 8, …, 2n} when n is even; FI(C_{2n}) = {2, 6, 10, …, 2n} when n is odd; and FI(C_{2n+1}) = {1, 3, 5, …, 2n − 1}. Elumalai [297] defines a cycle with a full set of cords as the graph P C_{n} obtained from C_{n} = v_{0}, v_{1}, v_{2}, …, v_{n−1} by adding the cords v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{(n−2)/2}, v_{(n+2)/2} when n is even and v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{(n−3)/2}, v_{(n+3)/2} when n is odd. Lee and Ng [551] prove: FI(P C_{2m+1}) = {3m − 2, 3m − 4, 3m − 6, …, 0} when m is even and FI(P C_{2m+1}) = {3m − 2, 3m − 4, 3m − 6, …, 1} when m is odd; FI(P C_{4}) = {1, 3}; for m ≥ 3, FI(P C_{2m}) = {3m − 5, 3m − 7, 3m − 9, …, 1} when m is even; FI(P C_{2m}) = {3m − 5, 3m − 7, 3m − 9, …, 0} when m is odd.

Salehi and Lee [739] determined the friendly index for various classes of trees. Among their results are: for a tree with q edges that has a perfect matching, the friendly index is the odd integers from 1 to q and for n ≥ 2, FI(P_{n})= {n − 1 − 2i| 0 ≤ i⌊(n − 1)/2⌋. Lee and Ng [551] define P C(n, p) as the graph obtained from the cycle C_{n} with consecutive vertices v_{0}, v_{1}, v_{2}, …, v_{n−1} by adding the p cords joining v_{i} to v_{n−i} for 1 ≤ p⌊n/2⌋ − 1. They prove FI(P C(2m + 1, p)) = {2m + p − 1, 2m + p − 3, 2m + p − 5, …, 1} if p is even and FI(P C(2m + 1, p)) = {2m + p − 1, 2m + p − 3, 2m + p − 5, …, 0} if p is odd; FI(P C(2m, 1)) = {2m − 1, 2m − 3, 2m − 5, …, 1}; for m ≥ 3, and p ≥ 2, FI(P C(2m, p)) = {2m + p − 4, 2m + p − 6, 2m + p − 8, …, 0} when p is even, and FI(P C(2m, p)) = {2m + p − 4, 2m + p − 6, 2m + p − 8, …, 1} when p is odd. More generally, they show that the integers in the friendly index of a cycle with an arbitrary nonempty set of parallel chords form an arithmetic progression with a common difference 2. Shiu and Kwong [811] determine the friendly index of the grids P_{n} × P_{2}.

In [552] Lee and Ng prove: for n ≥ 2, FI(C_{2n}×P_{2}) = {0, 4, 8, …, 6n−8, 6n} if n is even and FI(C_{2n} ×P_{2}) = {2, 6, 10, …, 6n−8, 6n} if n is odd; FI(C_{3} ×P_{2}) = {1, 3, 5}; for n ≥ 2, FI(C_{2m+1}×P_{2}) = {6n−1}∪{6n−5−2k| where k ≥ 0 and 6n−5−2k ≥ 0}; FI(M_{4n}) (here M_{4n} is the Möbius ladder with 4n steps) = {6n−4−4k| where k ≥ 0 and 6n−4−4k ≥ 0}; FI(M_{4n+2}) = {6n + 3} ∪ {6n − 5 − 2k| where k ≥ 0 and 6n − 5 − 2k > 0}. In [525] Kwong, Lee, and Ng completely determine the friendly index of 2-regular graphs with two components. As a corollary, they show that C_{m} ∪ C_{n} is cordial if and only if m+n = 0, 1 or 3 (mod 4). Ho, Lee, and Ng [436] determine the friendly index sets of stars and various regular windmills. In [950] Wen determines the friendly index of generalized wheels C_{n} + mK_{1} for all m > 1.

Lee and Ng [551] define a parallel chord of C_{n} as an edge of the form v_{i}v_{n−i} (i < n − 1) that is not an edge of C_{n}. For n ≥ 6, they call the cycle C_{n} with consecutive vertices v_{1}, v_{2}, …, v_{n} and the edges v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{(n−2)/2}v_{(n+2)/2} for n even and v_{2}v_{n−1}, v_{3}v_{n−2}, …, v_{(n−1)/2}v_{(n+3)/2} for n odd, C_{n} with a full set of parallel chords. They determine the friendly index of these graphs and show that for any cycle with an arbitrary non-empty set of parallel chords the numbers in its friendly index set form an arithmetic progression with common difference 2.

For a graph G(V, E) and a graph H rooted at one of its vertices v, Ho, Lee, and Ng [435] define a root-union of (H, v) by G as the graph obtained from G by replacing each vertex of G with a copy of the root vertex v of H to which is appended the rest of the structure of H. They investigate the friendly index set of the root-union of stars by cycles.

For a graph G(V, E), the total graph T (G) of G, is the graph with vertex set V ∪ E and edge set E ∪ {(v, uv)| v ∈ V, uv ∈ E}. Note that the total graph of the n-star is the friendship graph and the total graph of P_{n} is a triangular snake. Lee and Ng [548] use SP (1^{n}, m) to denote the spider with one central vertex joining n isolated vertices and a path of length m. They show: FI(K_{1} + 2nK_{2}) (friendship graph with 2n triangles) = {2n, 2n − 4, 2n − 8, …, 0} if n is even; {2n, 2n − 4, 2n − 8, …, 2} if n is odd; FI(K_{1} + (2n + 1)K_{2}) = {2n + 1, 2n − 1, 2n − 3, …, 1}; for n odd, FI(T (P_{n})) = {3n − 7, 3n − 11, 3n − 15, …, z} where z = 0 if n ≡ 1 (mod 4) and z = 2 if n ≡ 3 (mod 4); for n even, FI(T (P_{n})) = {3n−7, 3n−11, 3n−15, …, n+1}∪{n−1, n−3, n−5, …, 1}; for m ≤ n−1, for m+n even, FI(T (SP (1^{n}, m))) = {3(m+n)−4, 3(m+n)−8, 3(m+n)−12, …, (m+n) (mod 4)}; for m + n odd, FI(T (SP (1^{n}, m))) = {3(m + n) − 4, 3(m + n) − 8, 3(m + n) − 12, …, m + n + 2} ∪ {m + n, m + n − 2, m + n − 4, …, 1}; for n ≥ m and m + n even, FI(T (SP (1^{n}, m))) = {|4k − 3(m + n)| |(n − m + 2)/2 ≤ k ≤ m + n}; for n ≥ m and m + n odd, FI(T (SP (1^{n}, m))) = {|4k − 3(m + n)| |(n − m + 3)/2 ≤ k ≤ m + n}.

Kwong and Lee [523] determine the friendly index any number of copies of C_{3} that share an edge in common and the friendly index any number of copies of C_{4} that share an edge in common.

In [486] Kim, Lee, and Ng define the balance index set of a graph G as {|e_{0}(f) − e_{1}(f)| where f runs over all friendly labelings f of G}. Zhang, Lee, and Wen [528] investigate the balance index sets for the disjoint union of up to four stars and Zhang, Ho, Lee, and Wen [1006] investigate the balance index sets for trees with diameter at most four. Kwong, Lee, and Sarvate [526] determine the balance index sets for cycles with on pendant edge, flowers and regular windmills. Lee, Ng, and Tong [554] determine the balance index set of certain graphs obtained by starting with a copies of a given cycle and successively identifying one particular vertex of one with a particular vertex of the next. For graphs G and H and a bijection π from G to H, Lee and Su [575] define Perm(G, π, H) as the graph obtaining from the disjoint union of G and H by joining each v in G to π(v) with an edge. They determine the balanced index sets of the disjoint union of cycles and the balanced index sets for graphs of the form Perm(G, π, H) where G and H are regular graphs, stars, paths, and cycles with a chord. They conjecture that the balanced index set for every graph of the form Perm(G, π, H) is an arithmetic progression.

In [811] Shiu and Kwong define the full friendly index set of a graph G as {n_{0}(f) − n_{1}(f) where f runs over all friendly labelings f of G}. The full friendly index for P_{2} × P_{n} is given by Shiu and Kwong in [811] and upper and lowers bounds for the full friendly index of C_{m} × C_{n} are given by Shiu and Ling [821].

3.9 k-equitable Labelings

In 1990 Cahit [213] proposed the idea of distributing the vertex and edge labels among {0, 1, …, k − 1} as evenly as possible to obtain a generalization of graceful labelings as follows. For any graph G(V, E) and any positive integer k, assign vertex labels from {0, 1, …, k−1} so that when the edge labels induced by the absolute value of the difference of the vertex labels, the number of vertices labeled with i and the number of vertices labeled with j differ by at most one and the number of edges labeled with i and the number of edges labeled with j differ by at most one. Cahit has called a graph with such an assignment of labels k-equitable. Note that G(V, E) is graceful if and only if it is |E| + 1-equitable and G(V, E) is cordial if and only if it is 2-equitable. Cahit [212] has shown the following: C_{n} is 3-equitable if and only if n ≢ 3 (mod 6); the triangular snake with n blocks is 3-equitable if and only if n is even; the friendship graph C_{3}^{(n)} is 3-equitable if and only if n is even; an Eulerian graph with q ≡ 3 (mod 6) edges is not 3-equitable; and all caterpillars are 3-equitable [212]. Cahit [212] claimed to prove that W_{n} is 3-equitable if and only if n ≢ 3 (mod 6) but Youssef [996] proved that W_{n} is 3-equitable for all n ≥ 4. Youssef [994] also proved that if G is a k-equitable Eulerian graph with q edges and k ≡ 2 or 3 (mod 4) then q ≢ k (mod 2k). Cahit conjectures [212] that a triangular cactus with n blocks is 3-equitable if and only if n is even. In [213] Cahit proves that every tree with fewer than five end vertices has a 3-equitable labeling. He conjectures that all trees are k-equitable [214]. In 1999 Speyer and Szaniszló [870] proved Cahit’s conjecture for k = 3.

Bhut-Nayak and Telang have shown that crowns C_{n} ⊙ K_{1}, are k-equitable for k = n, …, 2n − 1 [172] and C_{n} ⊙ K_{1} is k-equitable for all n when k = 2, 3, 4, 5, and 6 [173].

In [758] Seoud and Abdel Maqsoud prove: a graph with n vertices and q edges in which every vertex has odd degree is not 3-equitable if n ≡ 0 (mod 3) and q ≡ 3 (mod 6); all fans except P_{2} + \overline{K_{1}} are 3-equitable; all double fans P_{n} + \overline{K_{2}} except P_{4} + \overline{K_{2}} are 3-equitable; P_{n}^{2} is 3-equitable for all n except 3; K_{1,1,n} is 3-equitable if and only if n ≡ 0 or 2 (mod 3); K_{1,2,n}, n ≥ 2, is 3-equitable if and only if n ≡ 2 (mod 3); K_{m,n}, 3 ≤ m ≤ n, is 3-equitable if and only if (m, n) = (4, 4); and K_{1,m,n}, 3 ≤ m ≤ n, is 3-equitable if and only if (m, n) = (3, 4).

Bapat and Limaye [129] have shown the following graphs are 3-equitable: helms H_{n}, n ≥ 4; flowers (see §2.2 for the definition); the one-point union of any number of helms; the one-point union of any number of copies of K_{4}; K_{4}-snakes (see §2.2 for the definition); C_{t}-snakes where t = 4 or 6; C_{5}-snakes where the number of blocks is not congruent to 3 modulo 6. A multiple shell MS{n_{1}^{t_{1}}, …, n_{r}^{t_{r}}} is a graph formed by t_{i} shells each of order n_{i}, 1 ≤ i ≤ r, that have a common apex. Bapat and Limaye [130] show that every multiple shell is 3-equitable.

Szaniszló [905] has proved the following: P_{n} is k-equitable for all k; K_{n} is 2-equitable if and only if n = 1, 2, or 3; K_{n} is not k-equitable for 3 ≤ k < n; S_{n} is k-equitable for all k; K_{2,n} is k-equitable if and only if n ≡ k − 1 (mod k), or n ≡ 0, 1, 2, …, ⌊k/2⌋ − 1 (mod k), or n = ⌊k/2⌋ and k is odd. She also proves that C_{n} is k-equitable if and only if k meets all of the following conditions: n ≠ k; if k ≡ 2, 3 (mod 4), then n ≠ k − 1; if k ≡ 2, 3 (mod 4) then n ≢ k (mod 2k).

Vickrey [923] has determined the k-equitability of complete multipartite graphs. He shows that for m ≥ 3 and k ≥ 3, K_{m,n} is k-equitable if and only if K_{m,n} is one of the following graphs: K_{4,4} for k = 3; K_{3,k−1} for all k; or K_{m,n} for k > mn. He also shows that when k is less than or equal to the number of edges in the graph and at least 3, the only complete multipartite graphs that are k-equitable are K_{kn+k−1,2,1} and K_{kn+k−1,1,1}. Partial results on the k-equitability of K_{m,n} were obtained by Krussel [519].

As a corollary of the result of Cairnie and Edwards [223] on the computational complexity of cordially labeling graphs it follows that the problem of finding k-equitable labelings of graphs is NP-complete as well.

Seoud and Abdel Maqsoud [759] call a graph k-balanced if the vertices can be labeled from {0, 1, …, k − 1} so that the number of edges labeled i and the number of edges labeled j induced by the absolute value of the differences of the vertex labels differ by at most 1. They prove that P_{n}^{2} is 3-balanced if and only if n = 2, 3, 4, or 6; for k ≥ 4, P_{n}^{2} is not k-balanced if k ≤ n − 2 or n + 1 ≤ k ≤ 2n − 3; for k ≥ 4, P_{n}^{2} is k-balanced if k ≥ 2n − 2; for k, m, n ≥ 3, K_{m,n} is k-balanced if and only if k ≥ mn; for m ≤ n, K_{1,m,n} is k-balanced if and only if (i) m = 1, n = 1 or 2, and k = 3; (ii) m = 1 and k = n + 1 or n + 2; or (iii) k ≥ (m + 1)(n + 1).

Bloom has used the term k-equitable to describe another kind of labeling (see [958] and [959]). He calls a graph k-equitable if the edge labels induced by the absolute value of the difference of the vertex labels have the property that every edge label occurs exactly k times. Bloom calls a graph of order n minimally k-equitable if the vertex labels are 1, 2,…, n and it is k-equitable. Both Bloom and Wojciechowski [958], [959] proved that C_{n} is minimally k-equitable if and only if k is a proper divisor of n. Barrientos and Hevia [144] proved that if G is k-equitable of size q = kw (in the sense of Bloom), then δ(G) ≤ w and Δ(G) ≤ 2w. Barrientos, Dejter, and Hevia [143] have shown that forests of even size are 2-equitable. They also prove that for k = 3 or k = 4 a forest of size kw is k-equitable if and only if its maximum degree is at most 2w and that if 3 divides mn+1, then the double star S_{m,n} is 3-equitable if and only if q/3 ≤ m ≤ ⌊(q − 1)/2⌋. (S_{m,n} is K_{2} with m pendant edges attached at one end and n pendant edges attached at the other end.) They discuss the k-equitability of forests for k ≥ 5 and characterize all caterpillars of diameter 2 that are k-equitable for all possible values of k. Acharya and Bhat-Nayak [29] have shown that coronas of the form C_{2n} ⊙ K_{1} are minimally 4-equitable. In [132] Barrientos proves that the one-point union of a cycle and a path (dragon) and the disjoint union of a cycle and a path are k-equitable for all k that divide the size of the graph. Barrientos and Havia [144] have shown the following: C_{n} × K_{2} is 2-equitable when n is even; books B_{n} (n ≥ 3) are 2-equitable when n is odd; the vertex union of k-equitable graphs is k-equitable; and wheels W_{n} are 2-equitable when n ≢ 3 (mod 4). They conjecture that W_{n} is 2-equitable when n ≡ 3 (mod 4) except when n = 3. Their 2-equitable labelings of C_{n} × K_{2} and the n-cube utilized graceful labelings of those graphs.

M. Acharya and Bhat-Nayak [30] have proved the following: the crowns C_{2n} ⊙ K_{1} are minimally 2-equitable, minimally 2n-equitable, minimally 4-equitable, and minimally n-equitable; the crowns C_{3n} ⊙ K_{1} are minimally 3-equitable, minimally 3n-equitable, minimally n-equitable, and minimally 6-equitable; the crowns C_{5n} ⊙ K_{1} are minimally 5-equitable, minimally 5n-equitable, minimally n-equitable, and minimally 10-equitable; the crowns C_{2n+1} ⊙ K_{1} are minimally (2n + 1)-equitable; and the graphs P_{kn+1} are kequitable.

In [134] Barrientos calls a k-equitable labeling optimal if the vertex labels are consecutive integers and complete if the induced edge labels are 1, 2, …, w where w is the number of distinct edge labels. Note that a graceful labeling is a complete 1-equitable labeling. Barrientos proves that C_{m} ⊙ nK_{1} (that is, an m-cycle with n pendant edges attached at each vertex) is optimal 2-equitable when m is even; C_{3} ⊙ nK_{1} is complete 2-equitable when n is odd; and that C_{3} ⊙ nK_{1} is complete 3-equitable for all n. He also shows that C_{n} ⊙ K_{1} is k-equitable for every proper divisor k of the size 2n. Barrientos and Havia [144] have shown that the n-cube (n ≥ 2) has a complete 2-equitable labeling and that K_{m,n} has a complete 2-equitable labeling when m or n is even. They conjecture that every tree of even size has an optimal 2-equitable labeling.

3.10 Hamming-graceful Labelings

Mollard, Payan, and Shixin [665] introduced a generalization of graceful graphs called Hamming-graceful. A graph G = (V, E) is called Hamming-graceful if there exists an injective labeling g from V to the set of binary |E|-tuples such that {d(g(v), g(u))| uv ∈ E} = {1, 2, …, |E|} where d is the Hamming distance. Shixin and Yu [829] have shown that all graceful graphs are Hamming-graceful; all trees are Hamming-graceful; C_{n} is Hamming-graceful if and only if n ≡ 0 or 3 (mod 4); if K_{n} is Hamming-graceful, then n has the form k^{2} or k^{2} + 2; and K_{n} is Hamming-graceful for n = 2, 3, 4, 6, 9, 11, 16, and 18. They conjecture that K_{n} is Hamming-graceful for n of the forms k^{2} and k^{2} + 2 for k ≥ 5.

4 Variations of Harmonious Labelings

4.1 Sequential and Strongly c-harmonious Labelings

Chang, Hsu, and Rogers [230] and Grace [376], [377] have investigated subclasses of harmonious graphs. Chang et al. define an injective labeling f of a graph G with q vertices to be strongly c-harmonious if the vertex labels are from {0, 1, …, q − 1} and the edge labels induced by f(x) + f(y) for each edge xy are c, …, c + q − 1. Grace called such a labeling sequential. In the case of a tree, Chang et al. modify the definition to permit exactly one vertex label to be assigned to two vertices whereas Grace allows the vertex labels to range from 0 to q with no vertex label being used twice. By taking the edge labels of a sequentially labeled graph with q edges modulo q, we obviously obtain a harmoniously labeled graph. It is not known if there is a graph that can be harmoniously labeled but not sequentially labeled. Grace [377] proved that caterpillars, caterpillars with a pendant edge, odd cycles with zero or more pendant edges, trees with α-labelings, wheels W_{2n+1}, and P_{n}^{2} are sequential. Liu and Zhang [619] finished off the crowns C_{2n} ⊙ K_{1}. (The case C_{2n+1}⊙K_{1} was a special case of Grace’s results. Liu [616] proved crowns are harmonious.) Bu [195] also proved that crowns are sequential as are all even cycles with m pendant edges attached at each vertex. Figueroa-Centeno, Ichishima, and Muntaner-Batle [324] proved that all cycles with m pendant edges attached at each vertex are sequential. Wu [963] has shown that caterpillars with m pendant edges attached at each vertex are sequential.

Singh has proved the following: C_{n}⊙K_{2} is sequential for all odd n > 1 [835]; C_{n}⊙P_{3} is sequential for all odd n [836]; K_{2}⊙C_{n} (each vertex of the cycle is joined by edges to the end points of a copy of K_{2}) is sequential for all odd n [836]; helms H_{n} are sequential when n is even [836]; and K_{1,n} +K_{2}, K_{1,n} +\overline{K}_{2}, and ladders are sequential [838]. Santhosh [744] has shown that C_{n} ⊙ P_{4} is sequential for all odd n ≥ 3. Both Grace [376] and Reid (see [349]) have found sequential labelings for the books B_{2n}. Jungreis and Reid [468] have shown the following graphs are sequential: P_{m} × P_{n} (m, n) ≠ (2, 2); C_{4m} × P_{n} (m, n) ≠ (1, 2); C_{4m+2} × P_{2n}; C_{2m+1} × P_{n}; and C_{4} × C_{2n} (n > 1). The graphs C_{4m+2} × C_{2n+1} and C_{2m+1} × C_{2n+1} fail to satisfy a necessary parity condition given by Graham and Sloane [380]. The remaining cases of C_{m}×P_{n} and C_{m}×C_{n} are open. Gallian, Prout, and Winters [350] proved that all graphs C_{n} × P_{2} with a vertex or an edge deleted are sequential.

Gnanajothi [367, pp. 68–78] has shown the following graphs are sequential: K_{1,m,n}; mC_{n}, the disjoint union of m copies of C_{n} if and only if m and n are odd; books with triangular pages or pentagonal pages; and books of the form B_{4n+1}, thereby answering a question and proving a conjecture of Gallian and Jungreis [349]. Sun [885] has also proved that B_{n} is sequential if and only if n ≢ 3 (mod 4).

Yuan and Zhu [1004] have shown that mC_{n} is sequential when m and n are odd. Although Graham and Sloane [380] proved that the Möbius ladder M_{3} is not harmonious, Gallian [345] established that all other Möbius ladders are sequential (see §2.3 for the definition of Möbius ladder). Chung, Hsu, and Rogers [230] have shown that K_{m,n} + K_{1}, which includes S_{m} + K_{1}, is sequential. Seoud and Youssef [768] proved that if G is sequential and has the same number of edges as vertices, then G + \overline{K_{n}} is sequential for all n.

Zhou [1012] has observed that for graphs other than trees, the graphs with k-sequential labelings coincide with the graphs with strongly k-harmonious labelings. Zhou and Yuan [1013] have shown that for every k-sequential graph G with p vertices and q edges and any positive integer m the graph (G + \overline{K_{m}}) + \overline{K_{n}} is also k-sequential when q − p + 1 ≤ m ≤ q − p + k. Zhou [1012] has shown that the analogous results hold for strongly kharmonious and strongly k-elegant graphs. Zhou and Yuan [1013] have shown that for every k-sequential graph G with p vertices and q edges and any positive integer m the graph (G + \overline{K_{m}}) + \overline{K_{n}} is k-sequential when q − p + 1 ≤ m ≤ q − p + k.

Shee [563] that every graph is a subgraph of a sequential graph. Lu [628] provides three techniques for constructing larger sequential graphs from some smaller one: an attaching construction, an adjoining construction and the join of two graphs. Using these, he obtains various families of sequential or strongly k-indexable graphs.

Singh and Varkey [841] call a graph with q edges odd sequential if the vertices can be labeled with distinct integers from the set {0, 1, 2, …, q} or, in the case of a tree from the set {0, 1, 2, …, 2q − 1}, such that the edge labels induced by addition of the labels of the endpoints take on the values {1, 3, 5, …, 2q − 1}. They prove that combs, grids, stars, and rooted trees of level 2 are odd sequential whereas odd cycles are not. Singh and Varkey call a graph G bisequential if both G and its line graph have a sequential labeling. They prove paths and cycles are bisequential.

Among the strongly 1-harmonious (also called strongly harmonious) graphs are: fans F_{n} with n ≥ 2 [230]; wheels W_{n} with n ≢ 2 (mod 3) [230]; K_{m,n} + K_{1} [230]; French windmills K_{4}^{(t)} [444], [473]; the friendship graphs C_{3}^{(n)} if and only if n ≡ 0 or 1 (mod 4) [444], [473], [976]; C_{4k}^{(t)} [886]; and helms [715].

Seoud, Diab, and Elsakhawi [761] have shown that the following graphs are strongly harmonious: K_{m,n} with an edge joining two vertices in the same partite set; K_{1,m,n}; the composition P_{n}[P_{2}] (see §2.3 for the definition); B(3, 2, m) and B(4, 3, m) for all m (see §2.4 for the notation); P_{n}^{2} (n ≥ 3); and P_{n}^{3} (n ≥ 3). Seoud et al. [761] have also proved: B_{2n} is strongly 2n-harmonious; P_{n} is strongly ⌊n/2⌋-harmonious; ladders L_{2k+1} are strongly (k + 1)-harmonious; and that if G is strongly c-harmonious and has an equal number of vertices and edges, then G + \overline{K_{n}} is also strongly c-harmonious.

Sethuraman and Selvaraju [798] have proved that the graph obtained by joining two complete bipartite graphs at one edge is graceful and strongly harmonious. They ask whether these results extend to any number of complete bipartite graphs.

For a graph G(V, E) Gayathri and Hemalatha [359] define an even sequential harmonious labeling f of G as an injection from V to {0, 1, 2, …, 2|E|} with the property that the induced mapping f^{+} from E to {2, 4, 6, …, 2|E|} defined by f^{+}(uv) = f(u) + f(v) when f(u) + f(v) is even, and f^{+}(uv) = f(u) + f(v) + 1 when f(u) + f(v) is odd, is an injection. They prove the following have even sequential harmonious labelings (all cases are the nontrivial ones): P_{n}, P_{n}^{+}, C_{n}(n ≥ 3), triangular snakes, quadrilateral snakes, Möbius ladders, P_{m} × P_{n}(m ≥ 2, n ≥ 2), K_{m,n}; crowns C_{m} ⊙ K_{1}, graphs obtained by joining the centers of two copies of K_{1,n} by a path; banana trees (see §2.1), P_{n}^{2}, closed helms (see §2.2), C_{3} ⊙ nK_{1}(n ≥ 2); D ⊙ K_{1,n} where D is a dragon (see §2.2); ⟨K_{1,n} : m⟩ (m, ngeq2) (see §4.5); the wreath product P_{n} ∗ \overline{K_{2}} (n ≥ 2) (see §4.5); combs P_{n} ⊙ K_{1}; the one-point union of the end point of a path to a vertex of a cycle (tadpole); the one-point union of the end point of a tadpole and the center of a star; the graphs P C_{n} obtained from C_{n} = v_{0}, v_{1}, v_{2}, …, v_{n−1} by adding the cords v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{(n−2)/2}, v_{(n+2)/2} when n is even and v_{1}v_{n−1}, v_{2}v_{n−2}, …, v_{(n−3)/2}, v_{(n+3)/2} when n is odd (that is, cycles with a full set of cords); P_{m} · nK_{1}; the one-point union of a vertex of a cycle and the center of a star; graphs obtained by joining the centers of two stars with an edge; graphs obtained by joining two disjoint cycles with an edge (dumbbells); graphs consisting of two even cycles of the same order sharing a common vertex with an arbitrary number of pendant edges attached at the common vertex (butterflys).

4.2 (k, d)-arithmetic Labelings

Acharya and Hegde [23] have generalized sequential labelings as follows. Let G be a graph with q edges and let k and d be positive integers. A labeling f of G is said to be (k, d)-arithmetic if the vertex labels are distinct nonnegative integers and the edge labels induced by f(x) + f(y) for each edge xy are k, k + d, k + 2d, …, k + (q − 1)d. They obtained a number of necessary conditions for various kinds of graphs to have a (k, d)-arithmetic labeling. The case where k = 1 and d = 1 was called additively graceful by Hegde [413]. Hegde [413] showed: K_{n} is additively graceful if and only if n = 2, 3, or 4; every additively graceful graph except K_{2} or K_{1,2} contains a triangle; and a unicyclic graph is additively graceful if and only if it is a 3-cycle or a 3-cycle with a single pendant edge attached. Jinnah and Singh [463] noted that P_{n}^{2} is additively graceful. Hegde [414] proved that if G is strongly k-indexable, then G and G + \overline{K_{n}} are (kd, d)-arithmetic. Acharya and Hegde [25] proved that K_{n} is (k, d)-arithmetic if and only if n ≥ 5 (see also [201]). They also proved that a graph with an α-labeling is a (k, d)-arithmetic for all k and d. Bu and Shi [201] proved that K_{m,n} is (k, d)-arithmetic when k is not of the form id for 1 ≤ i ≤ n − 1. For all d ≥ 1 and all r ≥ 0, Acharya and Hegde [23] showed the following: K_{m,n,1} is (d + 2r, d)-arithmetic; C_{4t+1} is (2dt + 2r, d)-arithmetic; C_{4t+2} is not (k, d)-arithmetic for any values of k and d; C_{4t+3} is ((2t + 1)d + 2r, d)-arithmetic; W_{4t+2} is (2dt + 2r, d)-arithmetic; and W_{4t} is ((2t + 1)d + 2r, d)-arithmetic. They conjecture that C_{4t+1} is (2dt + 2r, d)-arithmetic for some r and that C_{4t+3} is (2dt + d + 2r, d)-arithmetic for some r. Hegde and Shetty [427] proved the following: the generalized web W (t, n) (see §2.2 for the definition) is ((n − 1)d/2, d)-arithmetic and ((3n − 1)d/2, d)-arithmetic for odd n; the join of the generalized web W (t, n) with the center removed and \overline{K_{p}} where n is odd is ((n − 1)d/2, d)-arithmetic; every T_{p}-tree (see §3.2 for the definition) with q edges and every tree obtained by subdividing every edge of a T_{p}-tree exactly once is (k + (q − 1)d, d)-arithmetic for all k and d. Lu, Pan, and Li [630] proved that K_{1,m} ∪ K_{p,q} is (k, d)-arithmetic when k > (q − 1)d + 1 and d > 1.

Yu [1002] proved that a necessary condition for C_{4t+1} to be (k, d)-arithmetic is that k = 2dt + r for some r ≥ 0 and a necessary condition for C_{4t+3} to be (k, d)-arithmetic is that k = (2t + 1)d + 2r for some r ≥ 0. These conditions were conjectured by Acharya and Hegde [23]. Singh proved that the graph obtained by subdividing every edge of the ladder L_{n} is (5, 2)-arithmetic [834] and that the ladder L_{n} is (n, 1)-arithmetic [837]. He also proves that P_{m} × C_{n} is ((n − 1)/2, 1)-arithmetic when n is odd [837]. Acharya, Germina, and Anandavally [18] proved that the subdivision graph of the ladder L_{n} is (k, d)-arithmetic if either d does not divide k or k = rd for some r ≥ 2n and that P_{m} × P_{n} and the subdivision graph of the ladder L_{n} are (k, k)-arithmetic if and only if k is at least 3. Lu, Pan, and Li [630] proved that S_{m} ∪ K_{p,q} is (k, d)-arithmetic when k > (q − 1)d + 1 and d > 1.

A graph is called arithmetic if it is (k, d)-arithmetic for some k and d. Singh and Vilfred [843] showed that various classes of trees are arithmetic. Singh [837] has proved that the union of an arithmetic graph and an arithmetic bipartite graph is arithmetic. He conjectures that the union of arithmetic graphs is arithmetic. He provides an example to show that the converse is not true.

4.3 (k, d)-indexable Labelings

Acharya and Hegde [23] call a graph with p vertices and q edges (k, d)-indexable if there is an injective function from V to {0, 1, 2, …, p − 1} such that the set of edge labels induced by adding the vertex labels is a subset of {k, k+d, k+2d, …, k+q(d−1)}. When the set of edges is {k, k+d, k+2d, …, k+q(d−1)} the graph is said to be strongly (k, d)-indexable. A (k, 1)-graph is more simply called k-indexable and strongly 1-indexable graphs are simply called strongly indexable. Notice that strongly indexable graphs are a stronger form of sequential graphs and for trees and unicyclic graphs the notions of sequential labelings and strongly k-indexable labelings coincide. Zhou [1012] has shown that for every kindexable graph G with p vertices and q edges the graph (G + \overline{K_{q−p+k}}) + \overline{K_{1}} is strongly k-indexable. Acharaya and Hegde prove that the only nontrivial regular graphs that are strongly indexable are K_{2}, K_{3}, and K_{2} × K_{3}, and that every strongly indexable graph has exactly one nontrivial component that is either a star or has a triangle. Acharya and Hegde [23] call a graph with p vertices indexable if there is an injective labeling of the vertices with labels from {0, 1, 2, …, p − 1} such that the edge labels induced by addition of the vertex labels are distinct. They conjecture that all unicyclic graphs are indexable. This conjecture was proved by Arumugam and Germina [52] who also proved that all trees are indexable. Bu and Shi [202] also proved that all trees are indexable and that all unicyclic graphs with the cycle C_{3} are indexable. Hegde [414] has shown the following: every graph can be embedded as an induced subgraph of an indexable graph; if a connected graph with p vertices and q edges (q ≥ 2) is (k, d)-indexable, then d ≤ 2; P_{m} × P_{n} is indexable for all m and n; if G is a connected (1, 2)-indexable graph, then G is a tree; the minimum degree of any (k, 1)-indexable graph with at least two vertices is at most 3; a caterpillar with partite sets of orders a and b is strongly (1, 2)-indexable if and only if |a − b| ≤ 1; in a connected strongly k-indexable graph with p vertices and q edges, k ≤ p − 1; and if a graph with p vertices and q edges is (k, d)-indexable, then q ≤ (2p − 3 − k + d)/d. As a corollary of the latter, it follows that K_{n} (n ≥ 4) and wheels are not (k, d)-indexable.

Hegde and Shetty [427] proved that for n odd the generalized web graph W (t, n) with the center removed is strongly (n − 1)/2-indexable. Hegde and Shetty [431] define a level joined planar grid as follows. Let u be a vertex of P_{m} × P_{n} of degree 2. For every pair of distinct vertices v and w that do not have degree 4, introduce an edge between v and w provided that the distance from u to v equals the distance from u to w. They prove that every level joined planar grid is strongly indexable.

Section 5.2 of this survey includes a discussion of a labeling method called super edgemagic. In 2002 Hegde and Shetty [431] showed that a graph has a strongly k-indexable labeling if and only if it has a super edge-magic labeling.

4.4 Elegant Labelings

In 1981 Chang, Hsu, and Rogers [230] defined an elegant labeling f of a graph G with q edges as an injective function from the vertices of G to the set {0, 1, …, q} such that when each edge xy is assigned the label f(x) + f(y) (mod (q + 1)) the resulting edge labels are distinct and nonzero. Note that in contrast to the definition of a harmonious labeling, for an elegant labeling it is not necessary to make an exception for trees. Whereas the cycle C_{n} is harmonious if and only if n is odd, Chang et al. [230] proved that C_{n} is elegant when n ≡ 0 or 3 (mod 4) and not elegant when n ≡ 1 (mod 4). Chang et al. further showed that all fans are elegant and the paths P_{n} are elegant for n ≢ 0 (mod 4). Cahit [210] then showed that P_{4} is the only path that is not elegant. Balakrishnan, Selvam, and Yegnanarayanan [125] have proved numerous graphs are elegant. Among them are K_{m,n} and the mth-subdivision graph of K_{1,2n} for all m. They prove that the bistar B_{n,n} (K_{2} with n pendant edges at each endpoint) is elegant if and only if n is even. They also prove that every simple graph is a subgraph of an elegant graph and that several families of graphs are not elegant. Deb and Limaye [270] have shown that triangular snakes (see §2.2 for the definition) are elegant if and only if the number of triangles is not equal to 3 (mod 4). In the case where the number of triangles is 3 (mod 4) they show the triangular snakes satisfy a weaker condition they call semi-elegant whereby the edge label 0 is permitted. In [271] Deb and Limaye define a graph G with q edges to be near-elegant if there is an injective function f from the vertices of G to the set {0, 1, …, q} such that when each edge xy is assigned the label f(x)+f(y) (mod (q +1)) the resulting edge labels are distinct and not equal to q. Thus, in a near-elegant labeling, instead of 0 being the missing value in the edge labels, q is the missing value. Deb and Limaye show that triangular snakes where the number of triangles is 3 (mod 4) are near-elegant. For any positive integers α ≤ β ≤ γ where β is at least 2, the theta graph θ_{α,β,γ} consists of three edge disjoint paths of lengths α, β, and γ having the same end points. Deb and Limaye [271] provide elegant and nearelegant labelings for some theta graphs where α = 1, 2, or 3. Seoud and Elsakhawi [762] have proved that the following graphs are elegant: K_{1,m,n}; K_{1,1,m,n}; K_{2} + \overline{K_{m}}; K_{3} + \overline{K_{m}}; and K_{m,n} with an edge joining two vertices of the same partite set.

Sethuraman and Elumalai [781] have proved that for every graph G with p vertices and q edges the graph G + K_{1} + \overline{K_{m}} is graceful when m ≥ 2^{p} − p − q.

Sethuraman and Elumalai [781] proved that every graph is a vertex induced subgraph of a elegant graph and present an algorithm that permits one to start with any nontrivial connected graph and successively form supersubdivisions (see §2.7) that have a strong form of elegant labeling.

In [780] Sethuraman and Elumalai define a graph H to be a K_{1,m}-star extension of a graph G with p vertices and q edges at a vertex v of G where m > p − 1 − deg(v) if H is obtained from G by merging the center of the star K_{1,m} with v and merging p−1−deg(v) pendent vertices of K_{1,m} with the p − 1 − deg(v) nonadjacent vertices of v in G. They prove that for every graph G with p vertices and q edges and for every vertex v of G and every m ≥ 2^{p−1} − 1 − q, there is a K_{1,m}-star extension of G that is both graceful and harmonious. In the case where m ≥ 2^{p−1} − q, they show that G has a K_{1,m}-star extension that is elegant. Sethuraman and Selvaraju [799] have shown that certain cases of the union of any number of copies of K_{4} with one or more edges deleted and one edge in common are elegant.

Gallian extended the notion of harmoniousness to arbitrary finite Abelian groups as follows. Let G be a graph with q edges and H a finite Abelian group (under addition) of order q. Define G to be H-harmonious if there is an injection f from the vertices of G to H such that when each edge xy is assigned the label f(x) + f(y) the resulting edge labels are distinct. When G is a tree, one label may be used on exactly two vertices. Beals, Gallian, Headley, and Jungreis [148] have shown that if H is a finite Abelian group of order n > 1 then C_{n} is H-harmonious if and only if H has a non-cyclic or trivial Sylow 2-subgroup and H is not of the form Z_{2} × Z_{2} × ⋯ × Z_{2}. Thus, for example, C_{12} is not Z_{12}-harmonious but is (Z_{2} × Z_{2} × Z_{3})-harmonious. Analogously, the notion of an elegant graph can be extended to arbitrary finite Abelian groups. Let G be a graph with q edges and H a finite Abelian group (under addition) with q + 1 elements. We say G is H-elegant if there is an injection f from the vertices of G to H such that when each edge xy is assigned the label f(x) + f(y) the resulting set of edge labels is the non-identity elements of H. Beals et al. [148] proved that if H is a finite Abelian group of order n with n ≠ 1 and n ≠ 3, then C_{n−1} is H-elegant using only the non-identity elements of H as vertex labels if and only if H has either a non-cyclic or trivial Sylow 2-subgroup. This result completed a partial characterization of elegant cycles given by Chang, Hsu, and Rogers [230] by showing that C_{n} is elegant when n ≡ 2 (mod 4). Mollard and Payan [664] also proved that C_{n} is elegant when n ≡ 2 (mod 4) and gave another proof that P_{n} is elegant when n ≠ 4.

For a graph G(V, E) and an Abelian group H Valentin [920] defines a polychrome labeling of G by H to be a bijection f from V to H such that the edge labels induced by f(uv) = f(v) + f(u) are distinct. Valentin investigates the existence of polychrome labelings for paths and cycles for various Abelian groups.

4.5 Felicitous Labelings

Another generalization of harmonious labelings are felicitous labelings. An injective function f from the vertices of a graph G with q edges to the set {0, 1, …, q} is called felicitous if the edge labels induced by f(x) + f(y) (mod q) for each edge xy are distinct. (Recall a harmonious labeling only allows the vertex labels 0, 1, …, q − 1.) This definition first appeared in a paper by Lee, Schmeichel, and Shee in [563] and is attributed to E. Choo. Balakrishnan and Kumar [122] proved the conjecture of Lee, Schmeichel, and Shee [563] that every graph is a subgraph of a felicitous graph by showing the stronger result that every graph is a subgraph of a sequential graph. Among the graphs known to be felicitous are: C_{n} except when n ≡ 2 (mod 4) [563]; K_{m,n} when m, n > 1 [563]; P_{2} ∪ C_{2n+1} [563]; P_{2} ∪ C_{2n} [910]; P_{3} ∪ C_{2n+1} [563]; S_{m} ∪ C_{2n+1} [563]; K_{n} if and only if n ≤ 4 [782]; P_{n} + \overline{K_{m}} [782]; the friendship graph C_{3}^{(n)} for n odd [563]; P_{n} ∪ C_{3} [806]; P_{n} ∪ C_{n+3} [910]; and the one-point union of an odd cycle and a caterpillar [806]. Shee [802] conjectured that P_{m} ∪ C_{n} is felicitous when n > 2 and m > 3. Lee, Schmeichel, and Shee [563] ask for which m and n is the one-point union of n copies of C_{m} felicitous. They showed that in the case where mn is twice an odd integer the graph is not felicitous. In contrast to the situation for felicitous labelings, we remark that C_{4k} and K_{m,n} where m, n > 1 are not harmonious and the one-point union of an odd cycle and a caterpillar is not always harmonious. Lee, Schmeichel, and Shee [563] conjectured that the n-cube is felicitous. This conjecture was proved by Figueroa-Centeno and Ichishima in 2001 [320].

Balakrishnan, Selvam, and Yegnanarayanan [124] obtained numerous results on felicitous labelings. The wreath product, G∗H, of graphs G and H has vertex set V (G)×V (H) and (g_{1}, h_{1}) is adjacent to (g_{2}, h_{2}) whenever g_{1}g_{2} ∈ E(G) or g_{1} = g_{2} and h_{1}h_{2} ∈ E(H). They define H_{n,n} as the graph with vertex set {u_{1}, …, u_{n}; v_{1}, …, v_{n}} and edge set {u_{i}v_{j}| 1 ≤ i ≤ j ≤ n}. They let ⟨K_{1,n} : m⟩ denote the graph obtained by taking m disjoint copies of K_{1,n}, and joining a new vertex to the centers of the m copies of K_{1,n}. They prove the following are felicitous: H_{n,n}; P_{n} ∗ \overline{K_{2}}; ⟨K_{1,m} : m⟩; ⟨K_{1,2} : m⟩ when m ≢ 0 (mod 3), or m ≡ 3 (mod 6), or m ≡ 6 (mod 12); ⟨K_{1,2n} : m⟩ for all m and n ≥ 2; ⟨K_{1,2t+1} : 2n+1⟩ when n ≥ t; P_{n}^{k} when k = n−1 and n ≢ 2 (mod 4), or k = 2t and n ≥ 3 and k < n − 1; the join of a star and \overline{K_{n}}; and graphs obtained by joining two end vertices or two central vertices of stars with an edge. Yegnanarayanan [990] conjectures that the graphs obtained from an even cycle by attaching n new vertices to each vertex of the cycle is felicitous. This conjecture was verified by Figueroa-Centeno, Ichishima, and Muntaner-Batle in [324]. In [795] Sethuraman and Selvaraju [799] have shown that certain cases of the union of any number of copies of K_{4} with 3 edges deleted and one edge in common are felicitous. Sethuraman and Selvaraju [795] present an algorithm that permits one to start with any non-trivial connected graph and successively form supersubdivisions (see §2.7) that have a felicitous labeling. Krisha and Dulawat [517] give algorithms for finding graceful, harmonious, sequential, felicitous, and antimagic (see §5.7) labelings of paths.

Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] define a felicitous graph to be strongly felicitous if there exists an integer k so that for every edge uv, min{f(u), f(v)} ≤ k < max{f(u), f(v)}. For a graph with p vertices and q edges with q ≥ p − 1 they show that G is strongly felicitous if and only if G has an α-labeling (see §3.1). They also show that for graphs G_{1} and G_{2} with strongly felicitous labelings f_{1} and f_{2} the graph obtained from G_{1} and G_{2} by identifying the vertices u and v such that f_{1}(u) = 0 = f_{2}(v) is strongly felicitous and that the one-point union of two copies of C_{m} where m ≥ 4 and m is even is strongly felicitous. As a corollary they have that the one-point union n copies of C_{m} where m is even and at least 4 and n ≡ 2 (mod 4) is felicitous. They conjecture that the one-point union of n copies of C_{m} is felicitous if and only if mn ≡ 0, 1, or 3 (mod 4). In [329] Figueroa-Centeno, Ichishima, and Muntaner-Batle prove that 2C_{n} is strongly felicitous if and only if n is even and at least 4. They conjecture [329] that mC_{n} is felicitous if and only if mn ≢ 2 (mod 4) and that C_{m} ∪ C_{n} is felicitous if and only if m + n ≢ 2 (mod 4).

Chang, Hsu, and Rogers [230] have given a sequential counterpart to felicitous labelings. They call a graph with q edges strongly c-elegant if the vertex labels are from {0, 1, …, q} and the edge labels induced by addition are {c, c+1, …, c+q−1}. (A strongly 1-elegant labeling has also been called a consecutive labeling.) Notice that every strongly c-elegant graph is felicitous and that strongly c-elegant is the same as (c, 1)-arithmetic in the case where the vertex labels are from {0, 1, …, q}. Chang et al. [230] have shown: K_{n} is strongly 1-elegant if and only if n = 2, 3, 4; C_{n} is strongly 1-elegant if and only if n = 3; and a bipartite graph is strongly 1-elegant if and only if it is a star. Shee [803] has proved that K_{m,n} is strongly c-elegant for a particular value of c and obtained several more specialized results pertaining to graphs formed from complete bipartite graphs.

Seoud and Elsakhawi [764] have shown: K_{m,n} (m ≤ n) with an edge joining two vertices of the same partite set is strongly c-elegant for c = 1, 3, 5, …, 2n + 2; K_{1,m,n} is strongly celegant for c = 1, 3, 5, …, 2m when m = n, and for c = 1, 3, 5, …, m + n + 1 when m ≠ n; K_{1,1,m,m} is strongly c-elegant for c = 1, 3, 5, …, 2m+1; P_{n}+\overline{K_{m}} is strongly ⌊n/2⌋-elegant; C_{m} +\overline{K_{n}} is strongly c-elegant for odd m and all n for c = (m−1)/2, (m−1)/2+2, …, 2m when (m − 1)/2 is even and for c = (m − 1)/2, (m − 1)/2 + 2, …, 2m − (m − 1)/2 when (m − 1)/2 is odd; ladders L_{2k+1} (k > 1) are strongly (k + 1)-elegant; and B(3, 2, m) and B(4, 3, m) (see §2.4 for notation) are strongly 1-elegant and strongly 3-elegant for all m; the composition P_{n}[P_{2}] (see §2.3 for the definition) is strongly c-elegant for c = 1, 3, 5, …, 5n − 6 when n is odd and for c = 1, 3, 5, …, 5n − 5 when n is even; P_{n} is strongly ⌊n/2⌋-elegant; P_{n}^{2} is strongly c-elegant for c = 1, 3, 5, …, q where q is the number of edges of P_{n}^{2}; and P_{n}^{3} (n > 3) is strongly c-elegant for c = 1, 3, 5, …, 6k − 1 when n = 4k, c = 1, 3, 5, …, 6k + 1 when n = 4k + 1, c = 1, 3, 5, …, 6k + 3 when n = 4k + 2, c = 1, 3, 5, …, 6k + 5 when n = 4k + 3.

5 Magic-type Labelings

5.1 Magic Labelings

Motivated by the notion of magic squares in number theory, magic labelings were introduced by Sedláček [751] in 1963. Responding to a problem raised by Sedláček, Stewart [874] and [875] studied various ways to label the edges of a graph in the mid 1960s. Stewart calls a connected graph semi-magic if there is a labeling of the edges with integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v. (Berge [155] used the term “regularisable” for this notion.) A semi-magic labeling where the edges are labeled with distinct positive integers is called a magic labeling. Stewart calls a magic labeling supermagic if the set of edge labels consists of consecutive positive integers. The classic concept of an n × n magic square in number theory corresponds to a supermagic labeling of K_{n,n}. Stewart [874] proved the following: K_{n} is magic for n = 2 and all n ≥ 5; K_{n,n} is magic for all n ≥ 3; fans F_{n} are magic if and only if n is odd and n ≥ 3; wheels W_{n} are magic for n ≥ 4; and W_{n} with one spoke deleted is magic for n = 4 and for n ≥ 6. Stewart [874] also proved that K_{m,n} is semi-magic if and only if m = n. In [875] Stewart proved that K_{n} is supermagic for n ≥ 5 if and only if n > 5 and n ≢ 0 (mod 4). Sedláček [752] showed that Möbius ladders M_{n} (see §2.3 for the definition) are supermagic when n ≥ 3 and n is odd and that C_{n} × P_{2} is magic, but not supermagic, when n ≥ 4 and n is even. Shiu, Lam, and Lee [816] have proved: the composition of C_{m} and \overline{K}_{n} (see §2.3 for the definition) is supermagic when m ≥ 3 and n ≥ 2; the complete m-partite graph K_{n,n,...,n} is supermagic when n ≥ 3, m > 5 and m ≢ 0 (mod 4); and if G is an r-regular supermagic graph, then so is the composition of G and \overline{K}_{n} for n ≥ 3. Ho and Lee [434] showed that the composition of K_{m} and \overline{K_{n}} is supermagic for m = 3 or 5 and n = 2 or n odd. Bača, Holländer, and Lih [91] have found two families of 4-regular supermagic graphs. Shiu, Lam, and Cheng [813] proved that for n ≥ 2, mK_{n,n} is supermagic if and only if n is even or both m and n are odd. Ivanc̆o [450] gave a characterization of all supermagic regular complete multipartite graphs. He proved that Q_{n} is supermagic if and only if n = 1 or n is even and greater than 2 and that C_{n} × C_{n} and C_{2m} × C_{2n} are supermagic. He conjectures that C_{m} × C_{n} is supermagic for all m and n. Trenklér [914] has proved that a connected magic graph with p vertices and q edges other than P_{2} exits if and only if 5p/4 < q ≤ p(p − 1)/2. In [887] Sun, Guan, and Lee give an efficient algorithm for finding a magic labeling of a graph. In [952] Wen, Lee, and Sun show how to construct a supermagic multigraph from a given graph G by adding extra edges to G.

In [514] Kovář provides a general technique for constructing supermagic labelings of copies of certain kinds of regular supermagic graphs. In particular, he proves: if G is a supermagic r-regular graph (r ≥ 3) with a proper edge r coloring, then nG is supermagic when r is even and supermagic when r and n are odd; if G is a supermagic r-regular graph with m vertices and has a proper edge r coloring and H is a supermagic s-regular graph with n vertices and has a proper edge s coloring, then G × H is supermagic when r is even or n is odd and is supermagic when s or m is odd.

In [288] Drajnová, Ivančo, and Semaničová proved that the maximal number of edges in a supermagic graph of order n is 8 for n = 5 and \frac{n(n−1)}{2} for 6 ≤ n ≢ 0 (mod 4), and \frac{n(n−1)}{2} − 1 for 8 ≤ n ≡ 0 (mod 4). They also establish some bounds for the minimal number of edges in a supermagic graph of order n. Ivančo, and Semaničová [455] proved that every 3-regular triangle-free supermagic graph has an edge such that the graph obtained by contracting that edge is also supermagic and the graph obtained by contracting one of the edges joining the two n-cycles of C_{n} × K_{2} (n ≥ 3) is supermagic.

Sedláček [752] proves that graphs obtained from an odd cycle with consecutive vertices u_{1}, u_{2}, …, u_{m}, u_{m+1}, v_{m}, …, v_{1} (m ≥ 2) by joining each u_{i} to v_{i} and v_{i+1} and u_{1} to v_{m+1}, u_{m} to v_{1} and v_{1} to v_{m+1} are magic. Trenklér and Vetchý [917] have shown that if G has order at least 5, then G^{n} is magic for all n ≥ 3 and G^{2} is magic if and only if G is not P_{5} and G does not have a 1-factor whose every edge is incident with an end-vertex of G. Avadayappan, Jeyanthi, and Vasuki [53] have shown that k-sequential trees are magic (see §4.1 for the definition). Seoud and Abdel Maqsoud [758] proved that K_{1,m,n} is magic for all m and n and that P_{n}^{2} is magic for all n. However, Serverino has reported that P_{n}^{2} is not magic for n = 2, 3, and 5 [364]. Jeurissan [458] characterized magic connected bipartite graphs. Ivančo [451] proved that bipartite graphs with p ≥ 8 vertices, equal sized partite sets, and minimum degree greater than p are magic.

Characterizations of regular magic graphs were given by Doob [287] and necessary and sufficient conditions for a graph to be magic were given in [458], [460], and [278]. Some sufficient conditions for a graph to be magic are given in [285], [913], and [670]. The notion of magic graphs was generalized in [286] and [743].

Let m, n, a_{1}, a_{2}, …, a_{m} be positive integers where 1 ≤ a_{i} ≤ ⌊n/2⌋ and the a_{i} are distinct. The circulant graph C_{n}(a_{1}, a_{2}, …, a_{m}) is the graph with vertex set {v_{1}, v_{2}, …, v_{m}} and edge set {v_{i}v_{i+a_{j}} | 1 ≤ i ≤ n, 1 ≤ j ≤ m} where addition of indices is done modulo n. In [756] Semaničová characterizes magic circulant graphs and 3-regular supermagic circulant graphs. In particular, if G = C_{n}(a_{1}, a_{2}, …, a_{m}) has degree r at least 3 and d = gcd(a_{1}, n/2) then G is magic if and only if r = 3 and n/d ≡ 2 (mod 4), a_{1}/d ≡ 1 (mod 2) where d = gcd(a_{1}, n/2), or r ≥ 4 (a necessary condition for C_{n}(a_{1}, a_{2}, …, a_{m}) to be 3-regular is that n is even). In the 3-regular case, C_{n}(a_{1}, n/2) with d = gcd(a_{1}, n/2) G is supermagic if and only n/d ≡ 2 (mod 4), a_{1}/d ≡ 1 (mod 2) and d ≡ 1 (mod 2).

Semaničová also notes that a bipartite graph that is decomposable into an even number of Hamilton cycles is supermagic. As a corollary she obtains that C_{n}(a_{1}, a_{2}, …, a_{2k}) is supermagic in the case that n is even, every a_{i} is odd, and gcd(a_{2j−1}, a_{2j}, n) = 1 for i = 1, 2, …, 2k and j = 1, 2, …, k.

In [450] Ivančo completely determines the supermagic graphs that are the disjoint unions of complete k-partite graphs where every partite set has the same order.

Trenklér [915] extended the definition of supermagic graphs to include hypergraphs and proved that the complete k-uniform n-partite hypergraph is supermagic if n ≠ 2 or 6 and k ≥ 2 (see also [916]).

For connected graphs of size at least 5, Ivančo, Lastivkova, and Semaničová [454] provide a forbidden subgraph characterization of the line graphs that can be magic. As a corollary they obtain that the line graph of every connected graph with minimum degree at least 3 is magic. They also prove that the line graph of every bipartite regular graph of degree at least 3 is supermagic.

In 1976 Sedláček [752] defined a connected graph with at least two edges to be pseudomagic if there exists a real-valued function on the edges with the property that distinct edges have distinct values and the sum of the values assigned to all the edges incident to any vertex is the same for all vertices. Sedláček proved that when n ≥ 4 and n is even, the Möbius ladder M_{n} is not pseudo-magic and when m ≥ 3 and m is odd, C_{m} × P_{2} is not pseudo-magic.

Kong, Lee, and Sun [503] used the term “magic labeling” for a labeling of the edges with nonnegative integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v. In particular, the edge labels need not be distinct. They let M(G) denote the set of all such labelings of G. For any L in M(G), they let s(L) = max{L(e): e in E} and define the magic strength of G as m(G) = min{s(L): L in M(G)}. To distinguish these notions from others with the same names and notation, which we will introduced in the next section for labelings from the set of vertices and edges, we call the Kong, Lee, and Sun version the edge magic strength and use em(G) for min{s(L): L in M(G)} instead of m(G). Kong, Lee, and Sun [503] use DS(k) to denote the graph obtained by taking two copies of K_{1,k} and connecting the k pairs of corresponding leafs. They show: for k > 1, em(DS(k)) = k −1; em(P_{k} +K_{1}) is 1 for k = 1 or 2, k if k is even and greater than 2, and 0 if k is odd and greater than 1; for k ≥ 3, em(W (k)) = k/2 if k is even and em(W (k)) = (k − 1)/2 if k is odd; em(P_{2} × P_{2}) = 1, em(P_{2} × P_{n}) = 2 if n > 3, em(P_{m} × P_{n}) = 3 if m or n is even and greater than 2; em(C_{3}^{(n)}) = 1 if n = 1 (Dutch windmill – see §2.4) and em(C_{3}^{(n)}) = 2n − 1 if n > 1. They also prove that if G and H are magic graphs then G × H is magic and em(G × H) = max{em(G), em(H)} and that every connected graph is an induced subgraph of a magic graph (see also [305] and [322]). They conjecture that almost all connected graphs are not magic. In [560] Lee, Saba, and Sun show that the edge magic strength of P_{n}^{k} is 0 when k and n are both odd. Sun and Lee [888] show that the Cartesian, conjunctive, normal, lexicographic, and disjunctive products of two magic graphs are magic and the sum of two magic graphs is magic. They also determine the edge magic strengths of the products and sums in terms of the edge magic strengths of the components graphs.

S. M. Lee and colleagues [590] and [540] call a graph G k-magic if there is a labeling from the edges of G to the set {1, 2, …, k − 1} such that for each vertex v of G the sum of all edges incident with v is a constant independent of v. The set of all k for which G is k-magic is denoted by IM(G) and called the integer-magic spectrum of G. In [590] Lee and Wong investigate the integer-magic spectrum of powers of paths. They prove: IM(P_{4}^{2}) is {4, 6, 8, 10, …}; for n > 5, IM(P_{n}^{2}) is the set of all positive integers except 2; for all odd d > 1, IM(P_{2d}^{d}) is the set of all positive integers except 1; IM(P_{4}^{3}) is the set of all positive integers; for all odd n ≥ 5, IM(P_{n}^{3}) is the set of all positive integers except 1 and 2; and for all even n ≥ 6, IM(P_{n}^{3}) is the set of all positive integers except 2. For k > 3 they conjecture: IM(P_{n}^{k}) is the set of all positive integers when n = k + 1; the set of all positive integers except 1 and 2 when n and k are odd and n ≥ k; the set of all positive integers except 1 and 2 when n and k are even and k ≥ n/2; the set of all positive integers except 2 when n is even and k is odd and n ≥ k; and the set of all positive integers except 2 when n and k are even and k ≤ n/2.

In [540] Lee, Lee, Sun, and Wen investigated the integer-magic spectrum of various graphs such as stars, double stars (trees obtained by joining the centers of two disjoint stars K_{1,m} and K_{1,n} with an edge), wheels, and fans. In [738] Salehi and Bennett report that a number of the results of Lee et al. are incorrect and provide a detailed accounting of these errors as well as determine the integer-magic spectra of caterpillars.

Lee, Lee, Sun, and Wen [540] use the notation C_{m}@C_{n} to denote the graph obtained by starting with C_{m} and attaching paths P_{n} to C_{m} by identifying the endpoints of the paths with each successive pairs of vertices of C_{m}. They prove that IM(C_{m}@C_{n}) is the set of all positive integers if m or n is even and IM(C_{m}@C_{n}) is the set of all even positive integers if m and n are odd.

Lee, Valdés, and Ho [578] investigate the integer magic spectrum for special kinds of trees. For a given tree T they define the double tree DT of T as the graph obtained by creating a second copy T^{∗} of T and joining each end vertex of T to its corresponding vertex in T^{∗}. They prove that for any tree T, IM(DT) contains every positive integer with the possible exception of 2 and IM(DT) contains all positive integers if and only if the degree of every vertex that is not an end vertex is even. For a given tree T they define ADT, the abbreviated double tree of T, as the the graph obtained from DT by identifying the end vertices of T and T^{∗}. They prove that for every tree T, IM(ADT) contains every positive integer with the possible exceptions of 1 and 2 and IM(ADT) contains all positive integers if and only if T is a path.

Lee, Salehi, and Sun [562] have investigated the integer-magic spectra of trees with diameter at most four. Among their findings are: if n ≥ 3 and the prime power factorization of n−1 = p_{1}^{r_{1}}p_{2}^{r_{2}} ⋯ p_{k}^{r_{k}}, then IM(K_{1,n}) = p_{1}ℕ∪p_{2}ℕ∪⋯∪ p_{k}ℕ (here p_{i}ℕ means all positive integer multiples of p_{i}); for m, n ≥ 3, the double star IM(DS(m, m)) (that is, stars K_{m,1} and K_{n,1} that have an edge in common) is the set of all natural numbers excluding all divisors of m − 2 greater than 1; if the prime power factorization of m − n = p_{1}^{r_{1}}p_{2}^{r_{2}} ⋯ p_{k}^{r_{k}} and the prime power factorization of n − 2 = p_{1}^{s_{1}}p_{2}^{s_{2}} ⋯ p_{k}^{s_{k}}, (the exponents of the p_{i}s for m − n and n − 2 are permitted to be 0) then IM(DS(m, n)) = A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{k} where A_{i} = p_{i}^{1+s_{i}}ℕ if r_{i} > s_{i} ≥ 0 and A_{i} = ∅ if s_{i} ≥ r_{i} ≥ 0; for m, n ≥ 3, IM(DS(m, n)) = ∅ if and only if m − n divides n − 2; if m, n ≥ 3 and |m − n| = 1, then DS(m, n) is non-magic. Lee and Salehi [561] give formulas for the integer-magic spectra of trees of diameter four but they are too complicated to include here.

For a graph G(V, E) and a function f from the V to the positive integers, Salehi and Lee [740] define the functional extension of G by f, as the graph H with V (H) = ∪{u_{i}| u ∈ V (G) and i = 1, 2, …, f(u)} E(H) = ∪{u_{i}u_{j}| uv ∈ E(G), i = 1, 2, …, f(u); j = 1, 2, …, f(v)}. They determine the integer-magic spectra for P_{2}, P_{3} and P_{4}.

More specialized results about the integer-magic spectra of amalgamations of stars and cycles are given by Lee and Salehi in [561].

Table 4 summarizes the state of knowledge about magic-type labelings. In the table SM means semi-magic M means magic SPM means supermagic.

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová.

Table 4: Summary of Magic Labelings

Graph  Types  Notes
K_{n}  M  if n = 2, n ≥ 5 [874]
SPM  for n ≥ 5 iff n > 5
n ≢ 0 (mod 4) [875]
K_{m,n}  SM  if n ≥ 3 [874]
K_{n,n}  M  if n ≥ 3 [874]
fans f_{n}  M  iff n is odd, n ≥ 3 [874]
not SM if n ≥ 2 [364]
wheels W_{n}  M  if n ≥ 4 [874]
SM  if n = 5 or 6 [364]
wheels with one  M  if n = 4, n ≥ 6 [874]

spoke deleted

Möbius ladders M_{n}  SPM  if n ≥ 3, n is odd [752]
C_{n} × P_{2}  not SPM for n ≥ 4, n even [752]
C_{m}[\overline{K}_{n}]  SPM  if m ≥ 3, n ≥ 2 [816]
K\underbrace{n, n, …, n}_{p}  SPM  n ≥ 3, p > 5 and
p ≢ 0 (mod 4) [816]
composition of r-regular SPM  if n ≥ 3 [816]

SPM graph and \overline{K}_{n}

K_{k}[\overline{K}_{n}]  SPM  if k = 3 or 5, n = 2 or n odd [434]

Table 4: Summary of Magic Labelings continued

Graph  Types Notes
mK_{n,n}  SPM for n ≥ 2 iff n is even or

both n and m are odd [813]

Q_{n}  SPM iff n = 1 or n > 2 even [450]
C_{m} × C_{n}  SPM m = n or m and n are even [450]
C_{m} × C_{n}  SPM? for all m and n [450]
connected (p, q)-graph M  iff 5p/4 < q ≤ p(p − 1)/2 [914]

other than P_{2}

G^{i}  M  |G| ≥ 5, i ≥ 3 [917]
G^{2}  M  G ≠ P_{5} and G does not have a

1-factor whose every edge is incident with an end-vertex of G [917]

K_{1,m,n}  M  for all m, n [758]
P_{n}^{2}  M  for all n except 2, 3, 5 [758], [364]
G × H  M  iff G and H are magic [503]

5.2 Edge-magic Total and Super Edge-magic Labelings

In 1970 Kotzig and Rosa [509] defined a magic labeling of a graph G(V, E) as a bijection f from V ∪E to {1, 2, …, |V ∪E|} such that for all edges xy, f(x)+f(y)+f(xy) is constant (called the magic constant. To distinguish between this usage from that of Stewart we will call this labeling an edge-magic total labeling. (We note that for 2-regular graphs a vertex-magic total labeling is an edge-magic total labeling and vice versa.) Kotzig and Rosa proved: K_{m,n} has an edge-magic total labeling for all m and n; C_{n} has an edge-magic total labeling for all n ≥ 3 (see also [368], [730], [158], and [305]); and the disjoint union of n copies of P_{2} has an edge-magic total labeling if and only if n is odd. They further state that K_{n} has an edge-magic total labeling if and only if n = 1, 2, 3, 5 or 6 (see [510], [265], and [305]) and ask whether all trees have edge-magic total labelings. Wallis, Baskoro, Miller, and Slamin [943] enumerate every edge-magic total labeling of complete graphs. They also prove that the following graphs are edge-magic total: paths, crowns, complete bipartite graphs, and cycles with a single edge attached to one vertex. Enomoto, Llado, Nakamigana, and Ringel [305] prove that all complete bipartite graphs are edge-magic total. They also show that wheels W_{n} are not edge-magic total when n ≡ 3 (mod 4) and conjectured that all other wheels are edge-magic total. This conjecture was proved when n ≡ 0, 1 (mod 4) by Phillips, Rees, and Wallis [702] and when n ≡ 6 (mod 8) by Slamin, Bača, Lin, Miller, and Simanjuntak [847]. Fukuchi [344] verified all cases of the conjecture independently of the work of others. Slamin et al. further show that all fans are edge-magic total. Ringel and Llado [724] prove that a graph with p vertices and q edges is not edge-magic total if q is even and p + q ≡ 2 (mod 4) and each vertex has odd degree. Ringel and Llado conjecture that trees are edge-magic total. In [115] Baskar Babujee and Rao show that the path with n vertices has an edge-magic total labeling with magic constant (5n + 2)/2 when n is even and (5n + 1)/2 when n is odd. For stars with n vertices they provide an edge-magic total labeling with magic constant 3n. In [313] Eshghi and Azimi discuss a zero-one integer programming model for finding edge-magic total labelings of large graphs.

Santhosh [747] proved that for n odd and at least 3, the crown C_{n} ⊙ P_{2} has an edgemagic total labeling with magic constant (27n+3)/2 and for n odd and at least 3, C_{n}⊙P_{3} has an edge-magic total labeling with magic constant (39n + 3)/2.

Beardon [150] extended the notion of edge-magic total to countable infinite graphs G(V, E) (that is, V ∪ E is countable). His main result is that a countably infinite tree that processes an infinite simple path has a bijective edge-magic total labeling using the integers as labels. He asks whether all countably infinite trees have an edge-magic total labeling with the integers as labels and whether the graph with the integers as vertices and an edge joining every two distinct vertices has a bijective edge-magic total labeling using the integers.

Cavenagh, Combe, and Nelson [228] investigate edge-magic total labelings of countably infinite graphs with labels from a countable Abelian group A. Their main result is that if G is a countable graph that has an infinite set of mutually disjoint edges and A is isomorphic to a countable subgroup of the real numbers under addition then for any k in A there is an edge-magic labeling of G with elements from A that has magic constant k.

Balakrishnan and Kumar [122] proved that the join of \overline{K_{n}} and two disjoint copies of K_{2} is edge-magic total if and only if n = 3. Yegnanarayanan [991] has proved the following graphs have edge-magic total labelings: nP_{3} where n is odd; P_{n} + K_{1}; P_{n} × C_{3} (n ≥ 2); the crown C_{n} ⊙ K_{1}; and P_{m} × C_{3} with n pendant vertices attached to each vertex of the outermost C_{3}. He conjectures that for all n, C_{n} ⊙ \overline{K_{n}}, the n-cycle with n pendant vertices attached at each vertex of the cycle, and nP_{3} have edge-magic total labelings. In fact, Figueroa-Centeno, Ichishima, and Muntaner-Batle [329] have proved the stronger statement that for all n ≥ 3, the corona C_{n} ⊙ \overline{K_{m}} admits an edge-magic labeling where the set of vertex labels is {1, 2, …, |V |}. Yegnanarayanan [991] also introduces several variations of edge-magic labelings and provides some results about them. Kotzig [941] provides some necessary conditions for graphs with an even number of edges in which every vertex has odd degree to have an edge-magic total labeling. Craft and Tesar [265] proved that an r-regular graph with r odd and p ≡ 4 (mod 8) vertices can not be edgemagic total. Wallis [939] proved that if G is an edge-magic total r-regular graph with p vertices and q edges where r = 2^{t}s + 1 (t > 0) and q is even, then 2^{t+2} divides p.

Figueroa-Centeno, Ichishima, and Muntaner-Batle [323] have proved the following graphs are edge-magic total: P_{4} ∪ nK_{2} for n odd; P_{3} ∪ nK_{2}; P_{5} ∪ nK_{2}; nP_{i} for n odd and i = 3, 4, 5; 2P_{n}; P_{1} ∪ P_{2} ∪ ⋯ ∪ P_{n}; mK_{1,n}; C_{m} ⊙ nK_{1}; K_{1} ⊙ nK_{2} for n even; W_{2n}; K_{2} × \overline{K}_{n}, nK_{3} for n odd (the case nK_{3} for n even and larger than 2 is done in [643]); binary trees, generalized Petersen graphs (see also [683]), ladders (see also [954]), books, fans, and odd cycles with pendant edges attached to one vertex. Enomoto, Llado, Nakamigawa, and Ringel [305] conjecture that if G is a graph of order n+m that contains K_{n}, then G is not edge-magic total for n ≫ m. Wijaya and Baskoro [954] proved that P_{m} × C_{n} is edge-magic total for odd n at least 3. Ngurah and Baskoro [683] state that P_{2} × C_{n} is not edge-magic total. Hegde and Shetty [424] have shown that every T_{p}-tree (see §4.4 for the definition) is edge-magic total. Ngurah, Simanjuntak, and Baskoro [688] show that certain subdivisions of the star K_{1,3} have edge-magic and super edge-magic total labelings. Wallis [939] proves that a cycle with one pendent edge is edge-magic total. In [939] Wallis poses a large number of research problems about edge-magic total graphs.

In 1996 Erdős asked for M (n), the maximum number of edges that an edge-magic total graph of order n can have (see [265]). In 1999 Craft and Tesar [265] gave the bound ⌊n^{2}/4⌋ ≤ M (n) ≤ ⌊n(n − 1)/2⌋. For large n this was improved by Pikhurko [705] in 2006 to 2n^{2}/7 + O(n) ≤ M (n) ≤ (0.489 + ⋯ + o(1)n^{2}).

Avadayappan, Vasuki, and Jeyanthi [54] define the edge-magic total strength of a graph G as the minimum of all constants over all edge-magic total labelings of G. We denote this by emt(G). They use the notation < K_{1,n} : 2 > for the tree obtained from the bistar B_{n,n} (the graph obtained by joining the center vertices of two copies of K_{1,n} with an edge) by subdividing the edge joining the two stars. They prove: emt(P_{2n}) = 5n+1; emt(P_{2n+1}) = 5n + 3; emt(< K_{1,n} : 2 >) = 4n + 9; emt(B_{n,n}) = 5n + 6; emt((2n + 1)P_{2}) = 9n + 6; emt(C_{2n+1}) = 5n + 4; emt(C_{2n}) = 5n + 2; emt(K_{1,n}) = 2n + 4; emt(P_{n}^{2}) = 3n; and emt(K_{n,m}) ≤ (m + 2)(n + 1) where n ≤ m. Using an analogous definition for super edgemagic total strength, Swaninathan and Jeyanthi [902], [902], [903] provide results about the super edge-magic strength of trees, fire crackers, unicyclic graphs, and generalized theta graphs.

Hegde and Shetty [429] (see also [428]) define the maximum magic strength of a graph G as the maximum magic constant over all edge-magic total labelings of G. We use eMt(G) to denote the maximum magic strength of G. Hegde and Shetty call a graph G with p vertices strong magic if eMt(G) = emt(G); ideal magic if 1 ≤ eMt(G) − emt(G) ≤ p; and weak magic if eMt(G) − emt(G) > p. They prove that for an edge-magic total graph G with p vertices and q edges, eMt(G) = 3(p + q + 1) − emt(G). Using this result they obtain: P_{n} is ideal magic for n > 2; K_{1,1} is strong magic; K_{1,2} and K_{1,3} are ideal magic; and K_{1,n} is weak magic for n > 3; B_{n,n} is ideal magic; (2n + 1)P_{2} is strong magic; cycles are ideal magic; and the generalized web W (t, 3) (see §2.2 for the definition) with the central vertex deleted is weak magic.

Santhosh [747] has shown that for n odd and at least 3, eMt(C_{n} ⊙ P_{2}) = (27n + 3)/2 and for n odd and at least 3, (39n + 3)/2 ≤ eMt(C_{n} ⊙ P_{2}) ≤ (40n + 3)/2. Moreover, he proved that for n odd and at least 3 both C_{n} ⊙ P_{2} and C_{n} ⊙ P_{3} are weak magic. In [252] Chopra and Lee provide an number of families of super edge-magic graphs that are weak magic.

In [671] Murugan introduces the notions of almost-magic labeling, relaxed-magic labeling, almost-magic strength, and relaxed-magic strength of a graph. He determines the magic strength of Huffman trees and twigs of odd order and the almost-magic strength of nP_{2} (n is even) and twigs of even order. Also, he obtains a bound on the magic strength of the path-union P_{n}(m) and on the relaxed-magic strength of kS_{n} and kP_{n}.

Enomoto, Llado, Nakamigawa, and Ringel [305] call an edge-magic total labeling super edge-magic if the set of vertex labels is {1, 2, …, |V |} (Wallis [939] calls these labelings strongly edge-magic). They prove the following: C_{n} is super edge-magic if and only if n is odd; caterpillars are super edge-magic; K_{m,n} is super edge-magic if and only if m = 1 or n = 1; and K_{n} is super edge-magic if and only if n = 1, 2, or 3. They also prove that if a graph with p vertices and q edges is super edge-magic then, q ≤ 2p − 3. In [639] MacDougall and Wallis study super edge-magic (p, q)-graphs where q = 2p − 3. Enomoto et al. [305] conjecture that every tree is super edge-magic. Lee and Shan [571] have verified this conjecture for trees with up to 17 vertices with a computer. Kotzig and Rosa’s ([509] and [510]) proof that nK_{2} is edge-magic total when n is odd actually shows that it is super edge-magic. Kotzig and Rosa also prove that every caterpillar is super-edge magic. Figueroa-Centeno, Ichishima, and Muntaner-Batle prove the following: if G is a bipartite or tripartite (super) edge-magic graph, then nG is (super) edge-magic when n is odd [326]; if m is a multiple of n + 1, then K_{1,m} ∪ K_{1,n} is super edge-magic [326]; K_{1,2} ∪ K_{1,n} is super edge-magic if and only if n is a multiple of 3; K_{1,m} ∪ K_{1,n} is edge-magic if and only if mn is even [326]; K_{1,3} ∪ K_{1,n} is super edge-magic if and only if n is a multiple of 4 [326]; P_{m} ∪ K_{1,n} is super edge-magic when m ≥ 4 [326]; 2P_{n} is super edge-magic if and only if n is not 2 or 3; K_{1,m} ∪ 2nK_{2} is super edge-magic for all m and n [326]; C_{3} ∪ C_{n} is super edge-magic if and only if n ≥ 6 and n is even [329] (see also [382]); C_{4} ∪C_{n} is super edge-magic if and only if n ≥ 5 and n is odd [329] (see also [382]); C_{5} ∪C_{n} is super edge-magic if and only if n ≥ 4 and n is even [329]; if m is even and at least 6 and n is odd and satisfies n ≥ m/2 + 2, then C_{m} ∪ C_{n} is super edge-magic [329]; C_{4} ∪ P_{n} is super edge-magic if and only if n ≠ 3 [329]; C_{5} ∪ P_{n} is super edge-magic if n ≥ 4 [329]; if m is even and at least 6 and n ≥ m/2 + 2, then C_{m} ∪ P_{n} is super edge-magic [329]; and P_{m} ∪ P_{n} is super edge-magic if and only if (m, n) ≠ (2, 2) or (3,3) [329]. They [326] conjecture that K_{1,m} ∪ K_{1,n} is super edge-magic only when m is a multiple of n + 1 and they prove that if G is a super edge-magic graph with p vertices and q edges with p ≥ 4 and q ≥ 2p − 4, then G contains triangles. In [329] Figueroa-Centeno et al. conjecture that C_{m} ∪ C_{n} is super edge-magic if and only if m + n ≥ 9 and m + n is odd.

Avadayappan, Jeyanthi, and Vasuki [53] define the super magic strength of a graph G as sm(G) = min{s(L): L where L runs over all super edge-magic labelings of G}. They use the notation < K_{1,n} : 2 > for the tree obtained from the bistar B_{n,n} (the graph obtained by joining the center vertices of two copies of K_{1,n} with an edge) by subdividing the edge joining the two stars. They prove: sm(P_{2n}) = 5n + 1; sm(P_{2n+1}) = 5n + 3; sm(< K_{1,n} : 2 >) = 4n + 9; sm(B_{n,n}) = 5n + 6; sm((2n + 1)P_{2}) = 9n + 6; sm(C_{2n+1}) = 5n + 4; emt(C_{2n}) = 5n + 2; sm(K_{1,n}) = 2n + 4; and sm(P_{n}^{2}) = 3n. Note that in each case the super magic strength of the graph is the same as its magic strength.

Santhosh and Singh [746] proved that C_{n} ⊙ P_{2} and C_{n} ⊙ P_{3} are super edge-magic for all odd n ≥ 3 and prove for odd n ≥ 3, sm(C_{n} ⊙ P_{2}) = (15n + 3)/2 and (20n + 3) ≤ sm(C_{n} ⊙ P_{3}) ≤ (21n + 3)/2.

In his Ph.D. thesis [383] Gray proves that C_{3} ∪ C_{n} is super edge-magic if and only if n ≥ 6 and C_{4} ∪ C_{n} is super edge-magic if and only if n ≥ 5. His computer search shows that C_{5} ∪ 2C_{3} does not have a super edge-magic labeling.

In [939] Wallis posed the problem of investigating the edge-magic properties of C_{n} with the path of length t attached to one vertex. Kim and Park [487] call such a graph an (n, t)-kite. They prove that an (n, 1)-kite is super edge-magic if and only if n is odd and an (n, 3)-kite is super edge-magic if and only if n is odd and at least 5. Park, Choi, and Bae [697] show that (n, 2)-kite is super edge-magic if and only if n is even. Wallis [939] also posed the problem of determining when K_{2} ∪ C_{n} is super edge-magic. Park et al. [697] and [487] prove that K_{2} ∪ C_{n} is super edge-magic if and only if n is even. Kim and Park [487] show that the graph obtained by attaching a pendant edge to a vertex of degree one of a star is super-edge magic and that a super edge-magic graph with edge magic constant k and q edges satisfies q ≤ 2k/3 − 3.

Lee and Kong [537] use St(a_{1}, a_{2}, …, a_{n}) to denote the disjoint union of the n stars St(a_{1}), St(a_{2}), …, St(a_{n}). They prove the following graphs are super edge-magic: St(m, n) where n ≡ 0 mod(m+1); St(1, 1, n); St(1, 2, n); St(1, n, n); St(2, 2, n); St(2, 3, n); St(1, 1, 2, n) (n ≥ 2); St(1, 1, 3, n); St(1, 2, 2, n); and St(2, 2, 2, n). They conjecture that St(a_{1}, a_{2}, …, a_{n}) is super edge-magic when n > 1 is odd.

In [638] MacDougall and Wallis investigate the existence of super edge-magic labelings of cycles with a chord. They use C_{v}^{t} to denote the graph obtained from C_{v} by joining two vertices that are distance t apart in C_{v}. They prove: C_{4m+1}^{t} (m ≥ 3) has a super edge-magic labeling for every t except 4m − 4 and 4m − 8; C_{4m}^{t} (m ≥ 3) has a super edge-magic labeling when t ≡ 2 mod 4; and that C_{4m+2}^{t} (m > 1) has a super edge-magic labeling for all odd t other than 5, and for t = 2 and 6. They pose the problem of what values of t does C_{2n}^{t} have a super edge-magic labeling.

Enomoto, Masuda, and Nakamigawa [306] have proved that every graph can be embedded in a connected super edge-magic graph as an induced subgraph. Slamin, Bača, Lin, Miller, Simanjuntak [847] proved that the friendship graph consisting of n triangles is super edge-magic if and only if n is 3, 4, 5 or 7. Fukuchi proved [343] the generalized Petersen graph P (n, 2) (see §2.7 for the definition) is super edge-magic if n is odd and at least 3 while Xu, Yang, Xi, Haque, and Shen [974] showed that P (n, 3) is super edgemagic for odd n is odd and at least 5. Baskoro and Ngurah [146] showed that nP_{3} is super edge-magic for n ≥ 4 and n even.

Hegde and Shetty [431] showed that a graph is super edge-magic if and only if it is strongly k-indexable (see §4.1 for the definition). Figueroa-Centeno, Ichishima, and Muntaner-Batle [322] proved that a graph is super edge-magic if and only if it is strongly 1-harmonious and that every super edge-magic graph is cordial. They also proved that P_{n}^{2} and K_{2} × C_{2n+1} are super edge-magic. In [323] Figueroa-Centeno et al. show that the following graphs are super edge-magic: P_{3} ∪ kP_{2} for all k; kP_{n} when k is odd; k(P_{2} ∪ P_{n}) when k is odd and n = 3 or n = 4; and fans F_{n} if and only if n ≤ 6. They conjecture that kP_{2} is not super edge-magic when k is even. This conjecture has been proved by Z. Chen [245] who showed that kP_{2} is super edge-magic if and only if k is odd. Figueroa-Centeno et al. provide a strong necessary condition for a book to have a super edge-magic labeling and conjecture that for n ≥ 5 the book B_{n} is super edge-magic if and only if n is even or n ≡ 5 (mod 8). They prove that every tree with an α-labeling is super edge-magic. Yokomura (see [305]) has shown that P_{2m+1} × P_{2} and C_{2m+1} × P_{m} are super edge-magic (see also [322]). In [324], Figueroa-Centeno et al. proved that if G is a (super) edge-magic 2-regular graph, then G⊙\overline{K}_{n} is (super) edge-magic and that C_{m}⊙\overline{K}_{n} is super edge-magic. Fukuchi [342] shows how to recursively create super edge-magic trees from certain kinds of existing super edge-magic trees. Ngurah, Baskoro, and Simanjuntak [687] provide a method for constructing new (super) edge-magic graphs from existing ones. One of their results is that if G has an edge-magic total labeling and G has order p and size p or p − 1, then G ⊙ nK_{1} has an edge-magic total labeling.

Lee and Lee [539] investigate the existence of total edge-magic labelings and super edge-magic labelings of unicylic graphs. They obtain a variety of positive and negative results and conjecture that all unicyclic are edge-magic total.

Shiu and Lee [819] investigated edge labelings of multigraphs. Given a multigraph G with q edges they call a bijection from the set of edges of G to {1, 2, …, q} with the property that for each vertex v the sum of all edge labels incident to v is a constant independent of v a supermagic labeling of G. They use K_{2}[n] to denote the multigraph consisting of n edges joining 2 vertices and mK_{2}[n] to denote the disjoint union of m copies of K_{2}[n]. They prove that for m and n at least 2, mK_{2}[n] is supermagic if and only if n is even or if both m and n are odd.

In 1970 Kotzig and Rosa [509] defined the edge-magic deficiency, μ(G), of a graph G as the minimum n such that G ∪ nK_{1} is edge-magic total. If no such n exists they define μ(G) = ∞. In 1999 Figueroa-Centeno, Ichishima, and Muntaner-Batle [328] extended this notion to super edge-magic deficiency, μ_{s}(G), is the analogous way. They prove the following: μ_{s}(nK_{2}) = μ(nK_{2}) = n−1 (mod 2); μ_{s}(C_{n}) = 0 if n is odd; μ_{s}(C_{n}) = 1 if n ≡ 0 (mod 4); μ_{s}(C_{n}) = ∞ if n ≡ 2 (mod 4); μ_{s}(K_{n}) = ∞ if and only if n ≥ 5; μ_{s}(K_{m,n}) ≤ (m − 1)(n − 1); μ_{s}(K_{2,n}) = n − 1; and μ_{s}(F) is finite for all forests F. They also prove that if a graph G has q edges with q/2 odd, and every vertex is even, then μ_{s}(G) = ∞ and conjecture that μ_{s}(K_{m,n}) ≤ (m − 1)(n − 1). This conjecture was proved for m = 3, 4, and 5 by Hegde, Shetty, and Shankaran [432] using the notion of strongly k-indexable labelings.

In [327] Figueroa-Centeno, Ichishima, and Muntaner-Batle proved that μ_{s}(P_{m}∪K_{1,n}) = 1 if m = 2 and n is odd, or m = 3 and is not congruent to 0 mod 3, whereas in all other cases μ_{s}(P_{m} ∪ K_{1,n}) = 0. They also proved that μ_{s}(2K_{1,n}) = 1 when n is odd and μ_{s}(2K_{1,n}) ≤ 1 when n is even. They conjecture that μ_{s}(2K_{1,n}) = 1 in all cases. Other results in [327] are: μ_{s}(P_{m} ∪ P_{n}) = 1 when (m, n) = (2, 2) or (3, 3) and μ_{s}(P_{m} ∪ P_{n}) = 0 in all other cases; μ_{s}(K_{1,m} ∪ K_{1,n}) = 0 when mn is even and μ_{s}(K_{1,m} ∪ K_{1,n}) = 1 when mn is odd; μ(P_{m} ∪ K_{1,n}) = 1 when m = 2 and n is odd and μ(P_{m} ∪ K_{1,n}) = 0 in all other cases; μ(P_{m} ∪ P_{n}) = 1 when (m, n) = (2, 2) and μ(P_{m} ∪ P_{n}) = 0 in all other cases; μ_{s}(2C_{n}) = 1 when n is even and ∞ when n is odd; μ_{s}(3C_{n}) = 0 when n is odd; μ_{s}(3C_{n}) = 1 when n ≡ 0 (mod 4); μ_{s}(3C_{n}) = ∞ when n ≡ 2 (mod 4); and μ_{s}(4C_{n}) = 1 when n ≡ 0 (mod 4). They conjecture the following: μ_{s}(mC_{n}) = 0 when mn is odd; μ_{s}(mC_{n}) = 1 when mn ≡ 0 (mod 4); μ_{s}(mC_{n}) = ∞ when mn ≡ 2 (mod 4); μ_{s}(2K_{1,n}) = 1; and if F is a forest with two components, then μ(F) ≤ 1 and μ_{s}(F) ≤ 1. Santhosh and Singh [745] proved: for n odd at least 3, μ_{s}(K_{2} ⊙ C_{n}) ≤ (n − 3)/2; for n > 1, 1 ≤ μ_{s}(P_{n}[P_{2}]) = ⌈(n − 1)/2⌉; and for n ≥ 1, 1 ≤ μ_{s}(P_{n} × K_{4}) ≤ n.

A block of a graph is a maximal subgraph with no cut-vertex. The block-cut-vertex graph of a graph G is a graph H whose vertices are the blocks and cut-vertices in G; two vertices are adjacent in H if and only if one vertex is a block in G and the other is a cutvertex in G belonging to the block. A chain graph is a graph with blocks B_{1}, B_{2}, B_{3}, …, B_{k} such that for every i, B_{i} and B_{i+1} have a common vertex in such a way that the blockcut-vertex graph is a path. The chain graph with k blocks where each block is identical and isomorphic to the complete graph K_{n} is called the kK_{n}-path.

Ngurah, Baskoro, and Simanjuntak [686] investigate the exact values of μ_{s}(kK_{n}-path) when n = 2 or 4 for all values of k and when n = 3 for k ≡ 0, 1, 2 (mod 4), and give an upper bound for k ≡ 3 (mod 4). They determine the exact super edge-magic deficiencies for fans, double fans, wheels of small order and provide upper and lower bounds for the general case as well as bounds for some complete partite graphs. They also include some open problems. Lee and Wang [583] show that various chain graphs with blocks that are complete graphs are super edge-magic.

Z. Chen [245] has proved: the join of K_{1} with any subgraph of a star is super edgemagic; the join of two nontrivial graphs is super edge-magic if and only if at least one of them has exactly two vertices and their union has exactly one edge; and if a k-regular graph is super edge-magic, then k ≤ 3. Chen also obtained the following: there is a connected super edge-magic graph with p vertices and q edges if and only if p − 1 ≤ q ≤ 2p − 3; there is a connected 3-regular super edge-magic graph with p vertices if and only if p ≡ 2 (mod 4); and if G is a k-regular edge-magic total graph with p vertices and q edges then (p + q)(1 + p + q) ≡ 0 (mod 2d) where d = gcd(k − 1, q). As a corollary of the last result, Chen observes that nK_{2} + nK_{2} is not edge-magic total.

Another labeling that has been called “edge-magic” was introduced by Lee, Seah, and Tan in 1992 [569]. They defined a graph G = (V, E) to be edge-magic if there exists a bijection f: E → {1, 2, …, |E|} such that the induced mapping f^{+}: V → N defined by f^{+}(u) = ∑_{(u,v)∈E} f(u, v) (mod |V |) is a constant map. Lee conjectured that a cubic graph with p vertices is edge-magic if and only if p ≡ 2 (mod 4). Lee, Pigg, and Cox [557] verified this conjecture for prisms and several other classes of cubic graphs. They also show that C_{n} × K_{2} is edge-magic if and only if n is odd. Shiu and Lee [819] showed that the conjecture is not true for multigraphs and disconnected graphs. Lee, Seah, and Tan [569] establish that a necessary condition for a multigraph with p vertices and q edges to be edge-magic is that p divides q(q + 1) and they exhibit several new classes of cubic edge-magic graphs. They also proved: K_{n,n} (n ≥ 3) is edge-magic and K_{n} is edge-magic for n ≡ 1, 2 (mod 4) and for n ≡ 3 (mod 4) (n ≥ 7). Lee, Seah, and Tan further proved that following graphs are not edge-magic: all trees except P_{2}; all unicyclic graphs; and K_{n} where n ≡ 0 (mod 4). Schaffer and Lee [750] have proved that C_{m} × C_{n} is always edge-magic. Lee, Tong, and Seah [577] have conjectured that the total graph of a (p, p)-graph is edge-magic if and only if p is odd. They prove this conjecture for cycles. Lee, Kitagaki, Young, and Kocay [536] proved that a maximal outerplanar graph with p vertices is edge-magic if and only if p = 6. Shiu [810] used matrices with special properties to prove that the composition of P_{n} with \overline{K_{n}} and the composition of P_{n} with \overline{K_{kn}} where kn is odd and n is at least 3 have edge-magic labelings.

Chopra, Dios, and Lee [251] investigated the edge-magicness of joins of graphs. Among their results are: K(2, m) is edge-magic if and only if m = 4 or 10; the only possible edgemagic graphs of the form K(3, m) are those with m = 3, 5, 6, 15, 33, and 69; for any fixed m there are only finitely many n such that K(m, n) is edge-magic; for any fixed m there are only finitely many trees T such that T + \overline{K_{m}} is edge-magic; and wheels are not edge-magic.

For any graph G and any positive integer k the graph G[k], called the k-fold G, is the hypergraph obtained from G by replacing each edge of G with k parallel edges. Lee, Seah, and Tan [569] proved that for any graph G with p vertices, G[2p] is edge-magic and, if p is odd, G[p] is edge-magic. Shiu, Lam, and Lee [817] show that if G is an (n + 1, n)-multigraph, then G is edge-magic if and only if n is odd and G is isomorphic to the disjoint union of K_{2} and (n − 1)/2 copies of K_{2}[2]. They also prove that if G is a (2m + 1, 2m)-multigraph and k ≥ 2, then G[k] is edge-magic if and only if 2m + 1 divides k(k − 1). For a (2m, 2m − 1)-multigraph G and k at least 2, they show that G[k] is edge-magic if 4m divides (2m−1)k((2m−1)k +1) or if 4m divides (2m+k −1)k. In [815] Shiu, Lam, and Lee characterize the (p, p)-multigraphs that are edge-magic as mK_{2}[2] or the disjoint union of mK_{2}[2] and two particular multigraphs or the disjoint union of K_{2}, mK_{2}[2], and four particular multigraphs. They also show for every (2m + 1, 2m + 1)- multigraph G, G[k] is edge-magic for all k at least 2. Lee, Seah, and Tan [569] prove that the multigraph C_{n}[k] is edge-magic for k ≥ 2.

Tables 5 and 6 summarize what is known about edge-magic total labelings and super edge-magic total labelings. We use SEM to indicate the graphs have super edge-magic total labelings and EMT to indicate the graphs have edge-magic total labelings. A question mark following SEM or EMT indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová.

Table 5: Summary of Edge-magic Total Labelings

Graph  Types  Notes
P_{n}  EMT  [943]
trees  EMT?  [510], [724]
C_{n}  EMT  for n ≥ 3 [509], [368], [730], [158]
K_{n}  EMT  iff n = 1, 2, 3, 4, 5, or 6 [510], [265], [305]

enumeration of all EMT of K_{n} [943]

K_{m,n}  EMT  for all m and n [509]
crowns C_{n} ⊙ K_{1}  EMT  [943]
K_{m,n}  EMT  [943]
C_{n} with a single edge EMT  [943]

attached to one vertex

wheels W_{n}  EMT  iff n ≢ 3 (mod 4) [305],[343]
fans  EMT  [847], [322], [323]
(p, q)-graph  not EMT if q even and p + q ≡ 2 (mod 4) [724]

Table 5: Summary of Edge-magic Total continued

Graph  Types  Notes
nP_{2}  EMT  iff n odd [509]
P_{n} + K_{1}  EMT  [991]
P_{n} × C_{3}  EMT  n ≥ 2 [991]
crown C_{n} ⊙ K_{1}  EMT  [991]
r-regular graph  not EMT r odd and p ≡ 4 (mod 8) [265]
P_{3} ∪ nK_{2} and P_{5} ∪ nK_{2} EMT  [322], [323]
P_{4} ∪ nK_{2}  EMT  n odd [322], [323]
nP_{i}  EMT  n odd, i = 3, 4, 5 [991] [322],[323]
nP_{3}  EMT?  [991]
2P_{n}  EMT  [322], [323]
P_{1} ∪ P_{2} ∪ ⋯ ∪ P_{n}  EMT  [322], [323]
mK_{1,n}  EMT  [322], [323]
C_{m} ⊙ \overline{K}_{n}  EMT  [322], [323]
unicylic graphs  EMT?  [539]

Graph  Types  Notes
K_{1} ⊙ nK_{2}  EMT  n even [322], [323]
K_{2} × \overline{K}_{n}  EMT  [322], [323]
nK_{3}  EMT  iff n ≠ 2 odd [322], [323], [643]
binary trees  EMT  [322], [323]
P (m, n) (generalized  EMT  [322], [323], [683]

Petersen graph)

ladders  EMT  [322], [323]
books  EMT  [322], [323]
odd cycle with pendant edges EMT  [322], [323]

attached to one vertex

P_{m} × C_{n}  EMT  n odd n ≥ 3 [954]
P_{m} × P_{2}  EMT  m odd m ≥ 3 [954]
P_{2} × C_{n}  not EMT [683]
K_{1,m} ∪ K_{1,n}  EMT  iff mn is even [326]
G ⊙ \overline{K}_{n}  EMT  if G is EMT 2-regular graph [324]

Table 6: Summary of Super Edge-magic Labelings

Graph  Types Notes
C_{n}  SEM iff n is odd [305]
caterpillars  SEM [305], [509], [510]
K_{m,n}  SEM iff m = 1 or n = 1 [305]
K_{n}  SEM iff n = 1, 2 or 3 [305]
trees  SEM? [305]
nK_{2}  SEM iff n odd [245]
nG  SEM if G is a bipartite or tripartite

SEM graph and n odd [326]

K_{1,m} ∪ K_{1,n}  SEM if m is a multiple of n + 1 [326]
K_{1,m} ∪ K_{1,n}  SEM? iff m is a multiple of n + 1 [326]
K_{1,2} ∪ K_{1,n}  SEM iff n is a multiple of 3 [326]
K_{1,3} ∪ K_{1,n}  SEM iff n is a multiple of 4 [326]
P_{m} ∪ K_{1,n}  SEM if m ≥ 4 is even [326]
2P_{n}  SEM iff n is not 2 or 3 [326]
2P_{4n}  SEM for all n [326]
K_{1,m} ∪ 2nK_{1,2} SEM for all m and n [326]

Table 6: Summary of Super Edge-magic Labelings continued

Graph  Types Notes
C_{3} ∪ C_{n}  SEM iff n ≥ 6 even [329], [382]
C_{4} ∪ C_{n}  SEM iff n ≥ 5 odd [329], [382]
C_{5} ∪ C_{n}  SEM iff n ≥ 4 even [329]
C_{m} ∪ C_{n}  SEM if m ≥ 6 even and n odd n ≥ m/2 + 2 [329]
C_{m} ∪ C_{n}  SEM? iff m + n ≥ 9 and m + n odd [329]
C_{4} ∪ P_{n}  SEM iff n ≠ 3 [329]
C_{5} ∪ P_{n}  SEM if n ≠ 4 [329]
C_{m} ∪ P_{n}  SEM if m ≥ 6 even and n ≥ m/2 + 2 [329]
P_{m} ∪ P_{n}  SEM iff (m, n) ≠ (2, 2) or (3, 3) [329]
corona C_{n} ⊙ \overline{K}_{m} SEM n ≥ 3 [329]
St(m, n)  SEM n ≡ 0 (mod m + 1) [537]
St(1, k, n)  SEM k = 1, 2 or n [537]
St(2, k, n)  SEM k = 2, 3 [537]
St(1, 1, k, n)  SEM k = 2, 3 [537]
St(k, 2, 2, n)  SEM k = 1, 2 [537]
St(a_{1}, …, a_{n})  SEM? for n > 1 odd [537]

Graph  Types Notes
C_{4m}^{t}  SEM [638]
C_{4m+1}^{t}  SEM [638]
friendship graph  SEM iff n = 3, 4, 5, or 7 [847]

of n triangles

generalized Petersen  SEM if n ≥ 3 odd [342]

graph P (n, 2) (see §2.7)

nP_{3}  SEM if n ≥ 4 even [146]
P_{n}^{2}  SEM [322]
K_{2} × C_{2n+1}  SEM [322]
P_{3} ∪ kP_{2}  SEM for all k [323]
kP_{n}  SEM if k is odd [323]
k(P_{2} ∪ P_{n})  SEM if k is odd and n = 3, 4 [323]
fans F_{n}  SEM iff n ≤ 6 [323]
books B_{n}  SEM? iff n even or n ≡ 5 (mod 8)[323]
trees with α-labelings SEM [323]

Graph  Types  Notes
P_{2m+1} × P_{2}  SEM  [305], [322]
C_{2m+1} × P_{m}  SEM  [305], [322]
G ⊙ \overline{K}_{n}  SEM  if G is SEM 2-regular graph [324]
C_{m} ⊙ \overline{K}_{n}  SEM  [324]
join of K_{1} with any subgraph SEM  [245]

of a star

if G is k-regular SEM graph  then k ≤ 3 [245]
G is connected (p, q)-graph  SEM  G exists iff p − 1 ≤ q ≤ 2p − 3 [245]
G is connected 3-regular  SEM  iff p ≡ 2 (mod 4) [245]

graph on p vertices

nK_{2} + nK_{2}  not SEM [245]

5.3 Vertex-magic Total Labelings

MacDougall, Miller, Slamin, and Wallis [635] introduced the notion of a vertex-magic total labeling in 1999. For a graph G(V, E) an injective mapping f from V ∪ E to the set {1, 2, …, |V | + |E|} is a vertex-magic total labeling if there is a constant k, called the magic constant, such that for every vertex v, f(v) + ∑ f(vu) = k where the sum is over all vertices u adjacent to v (some authors use the term “vertex-magic” for this concept). They prove that the following graphs have vertex-magic total labelings: C_{n}; P_{n} (n > 2); K_{m,m} (m > 1); K_{m,m} − e (m > 2); and K_{n} for n odd. They also prove that when n > m + 1, K_{m,n} does not have a vertex-magic total labeling. They conjectured that K_{m,m+1} has a vertex-magic total labeling for all m and that K_{n} has vertex-magic total labeling for all n ≥ 3. The latter conjecture was proved by Lin and Miller [603] for the case that n is divisible by 4 while the remaining cases were done by MacDougall, Miller, Slamin, and Wallis [635]. Gray, MacDougall, and Wallis [392] then gave a simpler proof that all complete graphs are vertex-magic total. McQuillan and Smith [644] have shown that if n is odd, K_{n} has a vertex-magic total labeling with magic constant k if and only if (n/4)(n^{2} + 3) ≤ k ≤ (n/4)(n + 1)^{2}. In [643] McQuillan and McQuillan investigate the existence of vertex-magic labelings of nC_{3}. They prove: for every even integer n ≥ 4, nC_{3} is vertex-magic (and therefore also edge-magic); for each even integer n ≥ 6, nC_{3} has vertex-magic total labelings with at least 2n−2 different magic constants; if n ≡ 2 mod 4, two extra vertex-magic total labelings with the highest possible and lowest possible magic constants exist; if n = 2 · 3^{k}, k > 1, nC_{3} has a vertex-magic total labeling with magic constant k if and only if (1/2)(15n+4) ≤ k ≤ (1/2)(21n+2); if n is odd, there are vertexmagic total labelings for nC_{3} with n + 1 different magic constants. In [642] McQuillan provides a technique for constructing vertex-magic total labelings of 2-regular graphs. In particular, if m is an odd positive integer, G = C_{n_{1}} ∪ C_{n_{2}} ∪ ⋯ ∪ C_{n_{k}} has a vertex-magic total labeling, and J is any subset of I = {1, 2, …, k} then (∪_{i∈J} mC_{n_{i}}) ∪ (∪_{i∈I−J} mC_{n_{i}}) has a vertex-magic total labeling.

Lin and Miller [603] have shown that K_{m,m} is vertex-magic total for all m > 1 and that K_{n} is vertex-magic total for all n ≡ 0 (mod 4). Phillips, Rees, and Wallis [703] generalized the Lin and Miller result by proving that K_{m,n} is vertex-magic total if and only if m and n differ by at most 1. Cattell [226] has shown that a necessary condition for a graph of the form H + \overline{K_{n}} to be vertex-magic total is that the number of vertices of H is at least n − 1. As a corollary he gets that a necessary condition for K_{m_{1},m_{2},...,m_{r},n} where n is the largest size of any partite set to be vertex-magic total is that m_{1} + m_{2} + ⋯ + m_{r} ≥ n. He poses as an open question whether graphs that meet the conditions of the theorem are vertex-magic total. Cattell also proves that K_{1,n,n} has a vertex-magic total labeling when n is odd and K_{2,n,n} has a vertex-magic total labeling when n ≡ 3 (mod 4).

Miller, Bača, and MacDougall [655] have proved that the generalized Petersen graphs P (n, k) (see §2.7) for the definition) are vertex-magic total when n is even and k ≤ n/2−1. They conjecture that all P (n, k) are vertex-magic total when k ≤ (n−1)/2 and all prisms C_{n} × P_{2} are vertex-magic total. Bača, Miller, and Slamin [108] proved the first of these conjectures (see also [848] for partial results) while Slamin and Miller prove the second.

MacDougall et al. ([635], [637] and [390]) have shown: W_{n} has a vertex-magic total labeling if and only if n ≤ 11; fans F_{n} have a vertex-magic total labelings if and only if n ≤ 10; friendship graphs have vertex-magic total labelings if and only if the number of triangles is at most 3; K_{m,n} (m > 1) has a vertex-magic total labeling if and only if m and n differ by at most 1. Wallis [939] proved: if G and H have the same order and G ∪ H is vertex-magic total then so is G + H; if the disjoint union of stars is vertex-magic total, then the average size of the stars is less than 3; if a tree has n internal vertices and more than 2n leaves then it does not have a vertex-magic total labeling. Wallis [940] has shown that if G is a regular graph of even degree that has a vertex-magic total labeling then the graph consisting of an odd number of copies of G is vertex-magic total. He also proved that if G is a regular graph of odd degree (not K_{1}) that has a vertex-magic total labeling then the graph consisting of any number of copies of G is vertex-magic total.

Recall a helm H_{n} is obtained from a wheel W_{n} by attaching a pendant edge at each vertex of the n-cycle of the wheel. A generalized helm H(n, t) is a graph obtained from a wheel W_{n} by attaching a path on t vertices at each vertex of the n-cycle. A generalized web W(n, t) is a graph obtained from a generalized helm H(n, t) by joining the corresponding vertices of each path to form an n-cycle. Thus W(n, t) has (t + 1)n + 1 vertices and 2(t + 1)n edges. A generalized Jahangir graph J_{k,s} is a graph on ks + 1 vertices consisting of a cycle C_{ks} and one additional vertex that is adjacent to k vertices of C_{ks} at distance s to each other on C_{ks}. Rahim, Tomescu, and Slamin [714] prove: H_{n} has no vertex-magic total labeling for any n ≥ 3; W(n, t) has a vertex-magic total labeling for n = 3 or n = 4 and t = 1, but it is not vertex-magic total for n ≥ 17t + 12 and t ≥ 0; and J_{n,t+1} is vertex-magic total for n = 3 and t = 1, but it does not have this property for n ≥ 7t + 11 and t ≥ 1.

Fronček, Kovář, and Kovářová [331] proved that C_{n} × C_{2m+1} and K_{5} × C_{2n+1} are vertex-magic total. Kovář [512] furthermore proved some general results about products of certain regular vertex-magic total graphs. In particular, if G is a (2r + 1)-regular vertex-magic total graph that can be factored into an (r + 1)-regular graph and an rregular graph, then G × K_{5} and G × C_{n} for n even are vertex-magic total. He also proved that if G an r-regular vertex-magic total graph and H is a 2s-regular supermagic graph that can be factored into two s-regular factors, then their Cartesian product G × H is vertex-magic total if either r is odd, or r is even and |H| is odd.

Beardon [149] has shown that a necessary condition for a graph with c components, p vertices, q edges and a vertex of degree d to be vertex-magic total is (d+2)^{2} ≤ (7q^{2} +(6c+ 5)q +c^{2} +3c)/p. When the graph is connected this reduces to (d+2)^{2} ≤ (7q^{2} +11q +4)/p. As a corollary, the following are not vertex-magic total: wheels W_{n} when n ≥ 12; fans F_{n} when n ≥ 11; and friendship graphs C_{3}^{(n)} when n ≥ 4.

MacDougall has conjectured (see [513]) that every r-regular (r > 1) graph with the exception of 2K_{3} has a vertex-magic total labeling. As a corollary of a general result Kovář [513] has shown that every 2r-regular graph with an odd number of vertices and a Hamiltonian cycle has a vertex-magic total labeling.

Beardon [151] has investigated how vertices of small degree effect vertex-magic total labelings. Let G(p, q) be a graph with a vertex-magic total labeling with magic constant k and let d_{0} be the minimum degree of any vertex. He proves k ≤ (1 + d_{0})(p + q − d_{0}/2) and q < (1 + d_{0})q. He also shows that if G(p, q) is a vertex-magic graph with a vertex of degree one and t is the number of vertices of degree at least two, then t > q/3 ≥ (p−1)/3. Beardon [151] has shown that the graph obtained by attaching a pendant edge to K_{n} is vertex-magic total if and only if n = 2, 3, or 4.

Meissner and Zwierzyński [648] used finding vertex-magic total labelings of graphs as a way to compare the efficiency of parallel execution of a program versus sequential processing.

MacDougall, Miller, and Sugeng [636] define a super vertex-magic total labeling of a graph G(V, E) as a vertex-magic total labeling f of G with the additional property that f(V) = {1, 2, …, |V |} and f(E) = {|V | + 1, |V | + 2, …, |V | + |E|} (some authors use the term “super vertex-magic” for this concept). They show that a (p, q)-graph that has a super vertex-magic total labeling with magic constant k satisfies the following conditions: k = (p + q)(p + q + 1)/v − (v + 1)/2; k ≥ (41p + 21)/18; if G is connected, k ≥ (7p − 5)/2; p divides q(q + 1) if p is odd, and p divides 2q(q + 1) if p is even; if G has even order either p ≡ 0 (mod 8) and q ≡ 0 or 3 (mod 4) or p ≡ 4 (mod 8) and q ≡ 1 or 2 (mod 4); if G is r-regular and p and r have opposite parity then p ≡ 0 (mod 8) implies q ≡ 0 (mod 4) and p ≡ 4 (mod 8) implies q ≡ 2 (mod 4). They also show: C_{n} has a super vertex-magic total labeling if and only if n is odd; and no wheel, ladder, fan, friendship graph, complete bipartite graph or graph with a vertex of degree 1 has a super vertex-magic total labeling. They conjecture that no tree has a super vertex-magic total labeling and that K_{4n} has a super vertex-magic total labeling when n > 1. The latter conjecture was proved by Gómez in [372]. In [373] Gómez proved that if G is a d-regular graph that has a vertex-magic total labeling and k is a positive integer such that (k − 1)(d + 1) is even, then kG has a super vertex-magic total labeling. As a corollary, we have that if n and k are odd or if n ≡ 0 (mod 4) and n > 4, then kK_{n} has a super vertex-magic total labeling. Gómez also shows how graphs with super vertex-magic total labeling can be constructed from a given graph G with super vertex-magic total labeling by adding edges to G in various ways.

Swaminathan and Jeyanthi [900] prove the following graphs are super vertex-magic total: P_{n} if and only if n is odd and n ≥ 3; C_{n} if and only if n is odd; the star graph if and only if it is P_{2}; and mC_{n} if and only if m and n are odd. In [901] they prove the following: no super vertex-magic total graph has two or more isolated vertices or an isolated edge; a tree with n internal edges and tn leaves is not super vertex-magic total if t > (n + 1)/n; if Δ is the largest degree of any vertex in a tree T with p vertices and Δ > (−3 + \sqrt{1 + 16p})/2, then T is not super vertex-magic total; the graph obtained from a comb by appending a pendant edge to each vertex of degree 2 is super vertex-magic total ; the graph obtained by attaching a path with t edges to a vertex of an n-cycle is super vertex-magic total if and only if n + t is odd.

The Knödel graphs W_{Δ,n} with n even and degree Δ, where 1 ≤ Δ ≤ ⌊ log_{2}n⌋ have vertices pairs (i, j) with i = 1, 2 and 0 ≤ j ≤ n/2 − 1 where for every 0 ≤ j ≤ n/2 − 1 and there is an edge between vertex (1, j) and every vertex (2, (j + 2^{k} − 1) mod n/2), for k = 0, 1, …, Δ − 1. Xi, Yang, Mominul, and Wong [967] have shown that W_{3,n} is super vertex-magic total when n ≡ 0 mod 4.

For graphs G(V, E) and H Gutiérrez and Lladó [396] say that G is H-magic if there is a total labeling f from V ∪ E to {1, 2, …, |V | + |E|} such that for each subgraph H′ = (V′, E′) of G isomorphic to H, the sum Σf(v)+Σf(e) over all v in V′ and all e is E′ is constant. When f(V) is {1, 2, …, |V |}, G is called H-supermagic. They prove: K_{1,m} is K_{1,n}-supermagic for all 1 ≤ n ≤ m; for 1 < m < n, K_{m,n} is K_{1,t}-supermagic if and only if t = n; P_{m} is P_{n}-supermagic for 2 ≤ m ≤ n; and C_{m} is P_{n}-supermagic for 2 ≤ n < m where gcd(m, n(n − 1)) = 1. They also show how one can construct infinite families of H-magic graphs from a given graph H.

Balbuena, Barker, Das, Lin, Miller, Ryan and Slamin [118] call a vertex-magic total labeling of G(V, E) a strongly vertex-magic total labeling if the vertex labels are {1, 2, …, |V |}. They prove: the minimum degree of a strongly vertex-magic total graph is at least 2; for a strongly vertex-magic total graph G with n vertices and e edges, if 2e ≥ \sqrt{10n^{2} − 6n + 1} then the minimum degree of G is at least 3; and for a strongly vertex-magic total graph G with n vertices and e edges if 2e < \sqrt{10n^{2} − 6n + 1} then the minimum degree of G is at most 6. They also provide strongly vertex-magic total labelings for certain families of circulant graphs. In [642] McQuillan provides a technique for constructing vertex-magic total labelings of 2-regular graphs. In particular, if m is an odd positive integer, G = C_{n_{1}} ∪ C_{n_{2}} ∪ ⋯ ∪ C_{n_{k}} has a strongly vertex-magic total labeling, and J is any subset of I = {1, 2, …, k} then (∪_{i∈J} mC_{n_{i}}) ∪ (∪_{i∈I−J} mC_{n_{i}}) has a strongly vertex-magic total labeling.

Gray [383] proved that if G is a graph with a spanning subgraph H that possesses a strongly vertex-magic total labeling and G − E(H) is even regular, then G also possesses a strongly vertex-magic total labeling. As a corollary one has that regular Hamiltonian graphs of odd order has a strongly vertex-magic total labelings.

In a series of papers Gray and MacDougall expand on McQuillan’s technique to obtain variety of results. In [386] Gray and MacDougall show that for any r ≥ 4, every r-regular graph of odd order at most 17 has a strong vertex-magic total labeling. They also show that several large classes of r-regular graphs of even order, including some Hamiltonian graphs, have vertex-magic total labelings. They conjecture that every 2-regular graph of odd order possesses a strong vertex-magic total labeling if and only if it is not of the form (2t − 1)C_{3} ∪ C_{4} or 2tC_{3} ∪ C_{5}. They include five open problems.

In [387] Gray and MacDougall introduce a procedure called a mutation that transforms one vertex-magic totaling labeling into another one by swapping sets of edges among vertices that may result in different labeling of the same graph or a labeling of a different graph. Among their results are: a description of all possible mutations of a labeling of the path and the cycle; for all n ≥ 2 and all i from 1 to n − 1 the graphs obtained by identifying an end points of paths of lengths i, i + 1, and 2n − 2i − 1 have a vertex-magic total labeling; for odd n, the graph obtained by attaching an path of length n−m to an m cycle, (such graphs are called (m; n − m)-kites ) have strong vertex-magic total labelings for m = 3, …, n − 2; C_{2n+1} ∪ C_{4n+4} and 3C_{2n+1} have a strong vertex-magic total labeling; and for n ≥ 2, C_{4n} ∪C_{6n−1} have a strong vertex-magic total labeling. They conclude with three open problems.

Gray and MacDougall [388] show how to construct vertex-magic total labelings for several families of non-regular graphs, including the disjoint union of two other graphs already possessing vertex-magic total labelings. They prove that if G is a d-regular graph of order v and H a t-regular graph of order u each having a strong vertex magic total labeling and vd^{2} + 2d + 2v + 2u = 2tvd + 2t + ut^{2} then G ∪ H possesses a strong vertexmagic total labeling. They also provide bounds on the minimum degree of a graph with a vertex-magic total labeling.

In [389] Gray and MacDougall establish the existence of vertex-magic total labelings for several infinite classes of regular graphs. Their method enables them to begin with any even-regular graph and construct a cubic graph possessing a vertex-magic total labeling. that produces strong vertex-magic total labelings for many even order regular graphs. The construction also extends to certain families of non-regular graphs.

Balbuena, Barker, Lin, Miller and Sugeng [120] call vertex-magic total labeling an a-vertex consecutive magic labeling if the vertex labels are {a, a + 1, …, a + |V |}. For an a-vertex consecutive magic labeling of a graph G with p vertices and q edges they prove: if G has one isolated vertex, then a = q and (p − 1)^{2} + p^{2} = (2q + 1)^{2}; if q = p − 1, then p is odd and a = p − 1; if p = q, then p is odd and if G has minimum degree 1, then a = (p + 1)/2 or a = p; if G is 2-regular, then p is odd and a = 0 or p; and if G is r-regular, then p and r have opposite parities. They also define an b-edge consecutive magic labeling analogously and state some results for these labelings.

Wood [960] generalizes vertex-magic total and edge-magic total labelings by requiring only that the labels be positive integers rather than consecutive positive integers. He gives upper bounds for the minimum values of the magic constant and the largest label for complete graphs, forests, and arbitrary graphs.

Exoo, Ling, McSorley, Phillips, and Wallis [318] call a function λ a totally magic labeling of a graph G if λ is both an edge-magic total and a vertex-magic total labeling of G. A graph with such a labeling is called totally magic. Among their results are: P_{3} is the only connected totally magic graph that has a vertex of degree 1; the only totally magic graphs with a component K_{1} are K_{1} and K_{1} ∪ P_{3}; the only totally magic complete graphs are K_{1} and K_{3}; the only totally magic complete bipartite graph is K_{1,2}; nK_{3} is totally magic if and only if n is odd; P_{3} ∪ nK_{3} is totally magic if and only if n is even. In [942] Wallis asks: Is the graph K_{1,m} ∪ nK_{3} ever totally magic? That question was answered by Calhoun, Ferland, Lister, and Polhill [222] who proved that if K_{1,m} ∪ nK_{3} is totally magic then m = 2 and K_{1,2} ∪ nK_{3} is totally magic if and only if n is even.

McSorley and Wallis [646] examine the possible totally magic labelings of a union of an odd number of triangles and determine the spectrum of possible values for the sum of the label on a vertex and the labels on its incident edges and the sum of an edge label and the labels of the endpoints of the edge for all known totally magic graphs.

Gray and MacDougall [384] define an order n sparse semi-magic square to be an n × n array containing the entries 1, 2, …, m (for some m < n^{2}) once, has its remaining entries equal to 0, and whose rows and columns have a constant sum of k. They prove some basic properties of such squares and provide constructions for several infinite families of squares, including squares of all orders n ≥ 3. Moreover, they show how such arrays can be used to construct vertex-magic total labelings for certain families of graphs.

In Tables 7, 8 and 9 VMT means vertex-magic total labeling SVMT means super vertex magic total TM means totally magic labeling.

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2007.

Table 7: Summary of Vertex-magic Total Labelings

Graph  Labeling Notes
C_{n}  VMT  [635]
P_{n}  VMT  n > 2 [635]
K_{m,m} − e  VMT  m > 2 [635]
K_{m,n}  VMT  iff |m − n| ≤ 1 [703],[635],[637]
K_{n}  VMT  for n odd [635]
for n ≡ 2 (mod 4),n > 2 [603]
nK_{3}  VMT  iff n ≠ 2 [322], [323], [643]
mK_{n}  VMT  m ≥ 1, n ≥ 4 [645]
Petersen P (n, k) VMT  [108]
prisms C_{n} × P_{2} VMT  [848]
W_{n}  VMT  iff n ≤ 11 [635],[637]
F_{n}  VMT  iff n ≤ 10 [635],[637]

Table 7: Summary of Vertex-magic Total Labelings continued

Graph  Labeling Notes
friendship graphs  VMT  iff # of triangles ≤ 3 [635],[637]
G + H  VMT  |V (G)| = |V (H)|

and G ∪ H is VMT [939]

unions of stars  VMT  [939]
tree with n internal vertices not VMT [939]

and more than 2n leaves

nG VMT n odd, G regular of even

degree, VMT [940] G is regular of odd degree, VMT, but not K_{1} [940]

C_{n} × C_{2m+1}  VMT  [331]
K_{5} × C_{2n+1}  VMT  [331]
G × C_{2n}  VMT  G 2r + 1-regular VMT [512]
G × K_{5}  VMT  G 2r + 1-regular VMT [512]
G × H  VMT  G r-regular VMT, r odd

or r even and |H| odd,

H 2s-regular supermagic [512]

Table 8: Summary of Super Vertex-magic Total Labelings

Graph  Labeling  Notes
P_{n}  SVMT  iff n > 1 is odd [900]
C_{n}  SVMT  iff n is odd [900] and [636]
K_{1,n}  SVMT  iff n = 1 [900]
mC_{n}  SVMT  iff m and n are odd [900]
W_{n}  not SVMT [636]
ladders  not SVMT [636]
friendship graphs  not SVMT [636]
K_{m,n}  not SVMT [636]
dragons (see §2.2)  SVMT  iff order is even [901], [901]
Knödel graphs W_{3,n}  SVMT  n ≡ 0 (mod 4) [967]
graphs with minimum degree 1 not SVMT [636]
K_{4n}  SVMT  n > 1 [372]

Table 9: Summary of Totally Magic Labelings

Graph  Labeling Notes
P_{3} TM the only connected TM graph

with vertex of degree 1 [318]

K_{n}  TM  iff n = 1, 3 [318]
K_{m,n}  TM  iff K_{m,n} = K_{1,2} [318]
nK_{3}  TM  iff n is odd [318]
P_{3} ∪ nK_{3}  TM  iff n is even [318]
K_{1,m} ∪ nK_{3} TM  iff m = 2 and n is even [222]

5.4 Magic Labelings of Type (a, b, c)

A magic-type method for labeling the vertices, edges, and faces of a planar graph was introduced by Lih [600] in 1983. Lih defines a magic labeling of type (1,1,0) of a planar graph G(V, E) as an injective function from {1, 2, …, |V |+|E|} to V ∪E with the property that for each interior face the sum of the labels of the vertices and the edges surrounding that face is some fixed value. Similarly, Lih defines a magic labeling of type (1, 1, 1) of a planar graph G(V, E) with face set F as an injective function from {1, 2, …, |V |+|E|+|F |} to V ∪ E ∪ F with the property that for each interior face the sum of the labels of the face and the vertices and the edges surrounding that face is some fixed value. Lih calls a labeling involving the faces of a plane graph consecutive if for every integer s the weights of all s-sided faces constitute a set of consecutive integers. Lih gave consecutive magic labelings of type (1, 1, 0) for wheels, friendship graphs, prisms, and some members of the Platonic family. In [60] Bača shows that the cylinders C_{n} × P_{m} have magic labelings of type (1, 1, 0) when m ≥ 2, n ≥ 3, n ≠ 4. In [70] Bača proves that the generalized Petersen graph P (n, k) (see §2.7 for the definition)) has a consecutive magic labeling if and only if n is even and at least 4 and k ≤ n/2 − 1

Bača gave magic labelings of type (1, 1, 1) for fans [56], ladders [56], planar bipyramids (that is, 2-point suspensions of paths) [56], grids [63], hexagonal lattices [62], Möbius ladders [58], and P_{n} × P_{3} [59]. Kathiresan and Ganesan [480] show that the graph P_{a,b} consisting of b ≥ 2 internally disjoint paths of length a ≥ 2 with common end points has a magic labeling of type (1, 1, 1) when b is odd, and when a = 2 and b ≡ 0 (mod 4). They also show that P_{a,b} has a consecutive labeling of type (1, 1, 1) when b is even and a ≠ 2.

Bača [57], [66], [64], [59], [65] and Bača and Holländer [88] gave magic labelings of type (1, 1, 1) and type (1, 1, 0) for certain classes of convex polytopes. Kathiresan and Gokulakrishnan [482] provided magic labelings of type (1, 1, 1) for the families of planar graphs with 3-sided faces, 5-sided faces, 6-sided faces, and one external infinite face. Bača [61] also provides consecutive and magic labelings of type (0, 1, 1) (that is, an injective function from {1, 2, …, |E| + |F |} to E ∪ F with the property that for each interior face the sum of the labels of the face and the edges surrounding that face is some fixed value) and a consecutive labeling of type (1, 1, 1) for a kind of planar graph with hexagonal faces.

A magic labeling of type (1,0,0) of a planar graph G with vertex set V is an injective function from {1, 2, …, |V |} to V with the property that for each interior face the sum of the labels of the vertices surrounding that face is some fixed value. Kathiresan, Muthuvel, and Nagasubbu [483] define a lotus inside a circle as the graph obtained from the cycle with consecutive vertices a_{1}, a_{2}, …, a_{n} and the star with central vertex b_{0} and end vertices b_{1}, b_{2}, …, b_{n} by joining each b_{i} to a_{i} and a_{i+1} (a_{n+1} = a_{1}). They prove that these graphs (n ≥ 5) and subdivisions of ladders have consecutive labelings of type (1, 0, 0). Devaraj [281] proves that graphs obtained by subdividing each edge of a ladder exactly the same number of times has a magic labeling of type (1, 0, 0).

Bača, Baskoro, Jendroľ, and Miller [78] investigated various d-antimagic labelings for graphs in the shape of hexagonal honeycombs. They use H_{n}^{m} to denote the honeycomb graph with m rows, n columns, and mn 6-sided faces. They prove: for n odd H_{n}^{m}, has a 0-antimagic vertex labeling and a 2-antimagic edge labeling; if n is odd and mn > 1, H_{n}^{m} has a 1-antimagic face labeling; for n odd and mn > 1, H_{n}^{m} has d-antimagic labelings of type (1, 1, 1) for d = 1, 2, 3, and 4.

In Table 10 we use following abbreviations

M(a, b, c) magic labeling of type (a, b, c)

CM(a, b, c) consecutive magic labeling of type (a, b, c).

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová.

Table 10: Summary of Magic Labelings of Type (a, b, c)

Graph  Labeling Notes
W_{n}  CM(1,1,0) [600]
friendship graphs  CM(1,1,0) [600]
prisms  CM(1,1,0) [600]
cylinders C_{n} × P_{m}  M(1,1,0) m ≥ 2, n ≥ 3, n ≠ 4 [60]
fans F_{n}  M(1,1,1) [56]
ladders  M(1,1,1) [56]
planar bipyramids (see §5.3)  M(1,1,1) [56]
grids  M(1,1,1) [63]
hexagonal lattices  M(1,1,1) [62]
Möbius ladders  M(1,1,1) [58]
P_{n} × P_{3}  M(1,1,1) [59]
certain classes of  M(1,1,1) [57], [66], [64], [59]
convex polytopes  M(1,1,0) [65], [88]
certain classes of planar graphs M(0,1,1) [61]
with hexagonal faces  CM(0,1,1)

CM(1,1,1)

lotus inside a circle (see §5.3) CM(1,0,0) n ≥ 5 [483]
subdivisions of ladders  M(1,0,0) [281]
CM(1,0,0) [483]

5.5 Other Types of Magic Labelings

For any nontrivial Abelian group A under addition a graph G is said to be A-magic if there exists a labeling f of the edges of G with the nonzero elements of A such that the vertex labeling f^{+} defined by f^{+}(v) = Σf(vu) over all edges vu is a constant. In [871] and [872] Stanley noted that Z-magic graphs can be viewed in the more general context of linear homogeneous diophantine equations. Shiu, Lam, and Sun [818] have shown the following: the union of two edge-disjoint A-magic graphs with the same vertex set is A-magic; the Cartesian product of two A-magic graphs is A-magic; the lexicographic product of two A-magic connected graphs is A-magic; for an Abelian group A of even order a graph is A-magic if and only if the degrees of all of its vertices have the same parity; if G and H are connected and A-magic, G composed with H is A-magic; K_{m,n} is A-magic when m, n ≥ 2 and A has order at least 4; K_{n} with an edge deleted is A-magic when n ≥ 4 and A has order at least 4; all generalized theta graphs (§4.4 for the definition) are A-magic when A has order at least 4; C_{n} + \overline{K_{m}} is A-magic when n ≥ 3, m ≥ 2 and A has order at least 2; wheels are A-magic when A has order at least 4; flower graphs C_{m}@C_{n} are A-magic when m, n ≥ 2 and A has order at least 4 (C_{m}@C_{n} is obtained from C_{n} by joining the end points of a path of length m − 1 to each pair of consecutive vertices of C_{n}).

In [559] Lee, Saba, Salehi, and Sun investigate graphs that are A-magic where A = V_{4} ≈ Z_{2} ⊕ Z_{2} is the Klein four-group. Many of theorems are special cases of the results of Shiu, Lam, and Sun [818] given in the previous paragraph. They also prove the following are V_{4}-magic: a tree if and only if every vertex has odd degree; the star K_{1,n} if and only if n is odd; K_{m,n} for all m, n ≥ 2; K_{n} − e (edge deleted K_{n}) when n > 3; even cycles with k pendent edges if and only if k is even; odd cycles with k pendent edges if and only if k is odd; wheels; C_{n} + \overline{K_{2}}; generalized theta graphs; flowers graphs C_{m}@C_{n}; graphs that are copies of C_{n} that share a common edge; and G + \overline{K_{2}} whenever G is V_{4}-magic.

Low and Lee [624] have shown that if a graph is A_{1}-magic then it is A_{2}-magic for any subgroup A_{2} of A_{1} and for any nontrivial Abelian group A every Eulerian graph of even size is A-magic. For a connected graph G, Low and Lee define T (G) to be the graph obtained from G by adding a disjoint uv path of length 2 for every pair of adjacent vertices u and v. They prove that for every finite nontrivial Abelian group A the graphs T (P_{2k}) and T (K_{1,2n+1}) are A-magic. Shiu and Low [824] show that K_{k_{1},k_{2},...,k_{n}}(k_{i} ≥ 2) is A-magic, for all A where |A| ≥ 3. Lee, Salehi and Sun [562] have shown that for m, n ≥ 3 the double star DS(m, n) is Z-magic if and only if m = n.

In [522] Kwong and Lee call the set of all k for which a graph is Z_{k}-magic the integermagic spectrum of the graph. They investigate the integer-magic spectra of the coronas of some specific graphs including paths, cycles, complete graphs, and stars. Low and Sue [627] have obtained some results on the integer-magic spectra of tessellation graphs. Shiu and Low [825] provide the integer-magic spectra of sun graphs. Chopra and Lee [254] determined the integer-magic spectra of all graphs consisting of any number of pairwise disjoint paths with common end vertices (that is, generalized theta graphs). Low and Lee [624] show that Eulerian graphs of even size are A-magic for every finite nontrivial Abelian group A whereas Wen and Lee [951] provide two families of Eularian graphs that are not A-magic for every finite nontrivial Abelian group A and eight infinite families of Eulerian graphs of odd sizes that are A-magic for every finite nontrivial Abelian group A. Low and Lee [624] also prove that if A is an Abelian group and G and H are A-magic, then so are G × H and the lexicographic product of G and H. Low and Shiu [626] prove: K_{1,n} × K_{1,n} has a Z_{n+1}-magic labeling with magic constant 0; if G × H is Z_{2}-magic, then so are G and H; if G is Z_{m}-magic and H is Z_{n}-magic, then the integer-magic spectra of G × H contains all common multiples of m and n; if n is even and k_{i} ≥ 3 then the integer-magic spectra of P_{k_{1}} × P_{k_{2}} × ⋯ × P_{k_{n}} = {3, 4, 5, …}.

Shiu and Low [826] have introduced the notion of ring-magic as follows. Given a commutative ring R with unity, a graph G is called R-ring-magic if there exists a labeling f of the edges of G with the nonzero elements of R such that the vertex labeling f^{+} defined by f^{+}(v) = Σf(vu) over all edges vu and vertex labeling f^{×} defined by f^{×}(v) = Πf(vu) over all edges vu are constant. They give some results about R-ring-magic graphs.

In [218] Cahit says that a graph G(p, q) is total magic cordial (TMC) provided there is a mapping f from V (G) ∪ E(G) to {0, 1} such that (f(a) + f(b) + f(ab)) mod 2 is a constant modulo 2 for all edges ab ∈ E(G) and |f(0) − f(1)| ≤ 1 where f(0) denotes the sum of the number of vertices labeled with 0 and the number of edges labeled with 0 and f(1) denotes the sum of the number of vertices labeled with 1 and the number of edges labeled with 1. He says a graph G is total sequential cordial (TSC) if there is a mapping f from V (G) ∪ E(G) to {0, 1} such that for each edge e = ab with f(e) = |f(a) − f(b)| it is true that |f(0) − f(1)| ≤ 1 where f(0) denotes the sum of the number of vertices labeled with 0 and the number of edges labeled with 0 and f(1) denotes the sum of the number of vertices labeled with 1 and the number of edges labeled with 1. He proves that the following graphs have a TMC labeling: K_{m,n} (m, n > 1), trees, cordial graphs, and K_{n} if and only if n = 2, 3, 5, or 6. He also proves that the following graphs have a TSC labeling: trees; cycles; complete bipartite graphs; friendship graphs; cordial graphs; cubic graphs other than K_{4}; wheels W_{n} (n > 3); K_{4k+1} if and only if k ≥ 1 and \sqrt{k} is an integer; K_{4k+2} if and only if \sqrt{4k + 1} is an integer; K_{4k} if and only if \sqrt{4k + 1} is an integer; and K_{4k+3} if and only if \sqrt{k + 1} is an integer.

In 2001, Simanjuntak, Rodgers, and Miller [657] defined a 1-vertex magic vertex labeling of G(V, E) as a bijection from V to {1, 2, …, |V |} with the property that there is a constant k such that at any vertex v the sum ∑ f(u) taken over all neighbors of v is k. Among their results are: H × \overline{K}_{2k} has a 1-vertex-magic vertex labeling for any regular graph H; the symmetric complete multipartite graph with p parts, each of which contains n vertices, has a 1-vertex-magic vertex labeling if and only if whenever n is odd, p is also odd; P_{n} has a 1-vertex-magic vertex labeling if and only if n = 1 or 3; C_{n} has a 1-vertex-magic vertex labeling if and only if n = 4; K_{n} has a 1-vertex-magic vertex labeling if and only if n = 1; W_{n} has a 1-vertex-magic vertex labeling if and only if n = 4; a tree has a 1-vertex-magic vertex labeling if and only if it is P_{1} or P_{3}; and r-regular graphs with r odd do not have a 1-vertex-magic vertex labeling.

Balbuena, Barker, Lin, Miller, and Sugeng [126] call a vertex-magic total labeling of a graph G(V, E) an a-vertex consecutive magic labeling if the vertex labels are {a + 1, a + 2, …, a + |V |} where 0 ≤ a ≤ |E|. They prove: if a tree of order n has an a-vertex consecutive magic labeling then n is odd and a = n − 1; if G has an a-vertex consecutive magic labeling with n vertices and e = n edges, then n is odd and if G has minimum degree 1, then a = (n + 1)/2 or a = n; if G has an a-vertex consecutive magic labeling with n vertices and e edges such that 2a ≤ e and 2e ≥ \sqrt{6}n − 1, then the minimum degree of G is at least 2; if a 2-regular graph of order n has an a-vertex consecutive magic labeling, then n is odd and a = 0 or n; and if a r-regular graph of order n has an a-vertex consecutive magic labeling, then n and r have opposite parities.

Balbuena et al. also call a vertex-magic total labeling of a graph G(V, E) a b-edge consecutive magic labeling if the edge labels are {b+1, b+2, …, b+|E|} where 0 ≤ b ≤ |V |. They prove: if G has n vertices and e edges and has a b-edge consecutive magic labeling and one isolated vertex, then b = 0 and (n − 1)^{2} + n^{2} = (2e + 1)^{2}; if a tree with odd order has a b-edge consecutive magic labeling then b = 0; if a tree with even order has a b-edge consecutive magic labeling then it is P_{4}; a graph with n vertices and e edges such that e ≥ 7n/4 and b ≥ n/4 and a b-edge consecutive magic labeling has minimum degree 2; if a 2-regular graph of order n has a b-edge consecutive magic labeling, then n is odd and b = 0 or b = n; and if a r-regular graph of order n has an b-edge consecutive magic labeling, then n and r have opposite parities.

Sugeng and Miller [879] prove: If (V, E) has an a-vertex consecutive edge magic labeling, where a ≠ 0 and a ≠ |E|, then G is disconnected; if (V, E) has an a-vertex consecutive edge magic labeling, where a ≠ 0 and a ≠ |E|, then G cannot be the union of three trees with more than one vertex each; for each nonnegative a and each positive n, there is an a-vertex consecutive edge magic labeling with n vertices; the union of r stars and a set of r − 1 isolated vertices has an s-vertex consecutive edge magic labeling, where s is the minimum order of the stars; for every b every caterpillar has a b-edge consecutive edge magic labeling; if a connected graph G with n vertices has a b-edge consecutive edge magic labeling where 1 ≤ b ≤ n − 1, then G is a tree; the union of r stars and a set of r − 1 isolated vertices has an r-edge consecutive edge magic labeling.

In [622] Lladó and Moragas introduce the notion of cycle-magic labelings. Given a fixed cycle C, a graph G = (V, E) is said to be C-magic if every edge of G belongs to at least one subgraph isomorphic to C and there is a total labeling from V ∪ E to {1, 2, …, |V | + |E|} such that, for every subgraph H′ = (V′, E′) isomorphic to C, the sum of all vertex labels in V′ plus the sum of all edge labels in E′ is fixed constant. When the vertex labels are 1, 2, …, |V |, G is called C-supermagic. They show: wheels W_{n} with n odd are C_{3}-magic; the Cartesian product of a C_{4}-free supermagic graph with K_{2} is C_{4}-magic (in particular, for n odd, prisms C_{n} × K_{2} and books K_{1,n} × K_{2} are C_{4}- magic); windmills K_{r}^{(k)} (see §2.4 for the definition) are C_{r}-magic; and subdivided wheels and uniform Θ-graphs (i.e., graphs consisting of any number of internally disjoint paths of uniform length joined at two end points) are cycle-magic. Selvagopal and Jeyanthi [755] have shown that n-polygonal snakes are C_{n}-supermagic (see §2.2 for the definition).

6 Antimagic-type Labelings

6.1 Antimagic Labelings

Hartsfield and Ringel [405] introduced antimagic graphs in 1990. A graph with q edges is called antimagic if its edges can be labeled with 1, 2, …, q such that the sums of the labels of the edges incident to each vertex are distinct. Among the graphs they prove are antimagic are: P_{n} (n ≥ 3), cycles, wheels, and K_{n} (n ≥ 3). T. Wang [945] has shown that the toroidal grids C_{n_{1}} × C_{n_{2}} × ⋯ × C_{n_{k}} are antimagic and, more generally, graphs of the form G × C_{n} are antimagic if G is an r-regular antimagic graph with r > 1. Cheng [249] proved that all Cartesian products or two or more regular graphs of positive degree are antimagic and that if G is j-regular and H has maximum degree at most k, minimum degree at least one (G and H need not be connected), then G × H is antimagic provided that j is odd and j^{2} − j ≥ 2k, or j is even and j^{2} > 2k. Wang and Hsiao [946] prove the following graphs are antimagic: G × P_{n} (n > 1) where G is regular; G × K_{1,n} where G is regular; compositions G[H] (see §2.3 for the definition) where H is d-regular with d > 1; and the Cartesian product of any double star (see stars with an edge joining their centers) and a regular graph. In [248] Cheng proved that P_{n_{1}} × P_{n_{2}} × ⋯ × P_{n_{t}} (t ≥ 2) is antimagic.

Hartsfield and Ringel [405] conjecture that every tree except P_{2} is antimagic and, moreover, every connected graph except P_{2} is antimagic. Alon, Kaplan, Lev, Roditty, and Yuster [41] use probabilistic methods and analytic number theory to show that this conjecture is true for all graphs with n vertices and minimum degree Ω(log n). They also prove that if G is a graph with n ≥ 4 vertices and Δ(G) ≥ n − 2, then G is antimagic and all complete partite graphs except K_{2} are antimagic. Chawathe and Krishna [239] proved that every complete m-ary tree is antimagic.

Sonntag [865] has extended the notion of antimagic labelings to hypergraphs. He shows that certain classes of cacti, cycle, and wheel hypergraphs have antimagic labelings. In [102] Bača, MacDougall, Miller, Slamin and Wallis survey results on antimagic, edgemagic total, and vertex-magic total labelings.

Hefetz [412] calls a graph with q edges k-antimagic if its edges can be labeled with 1, 2, …, q + k such that the sums of the labels of the edges incident to each vertex are distinct. In particular, antimagic is the same as 0-antimagic. More generally, given a weight function ω from the vertices to the natural numbers Hefetz calls a graph with q edges (ω, k)-antimagic if its edges can be labeled with 1, 2, …, q + k such that the sums of the labels of the edges incident to each vertex and the weight assigned to each vertex by ω are distinct. In particular, antimagic is the same as (ω, 0)-antimagic where ω is the zero function. Using Alon’s combinatorial nullstellensatz [40] as his main tool, Hefetz has proved the following: a graph with 3^{m} vertices and a K_{3} factor is antimagic; a graph with q edges and at most one isolated vertex and no isolated edges is (ω, 2q − 4)-antimagic; a graph with p > 2 vertices that admits a 1-factor is (p − 2)-antimagic; a graph with p vertices and maximum degree n−k, where k ≥ 3 is any function of p is (3k −7)-antimagic and, in the case that p ≥ 6k^{2}, is (k − 1)-antimagic.

The concept of an (a, d)-antimagic labelings was introduced by Bodendiek and Walther [183] in 1993. A connected graph G = (V, E) is said to be (a, d)-antimagic if there exist positive integers a, d and a bijection f: E → {1, 2, …, |E|} such that the induced mapping g_{f}: V → N, defined by g_{f}(v) = ∑{f(uv)| uv ∈ E(G)}, is injective and g_{f}(V) = {a, a + d, …, a + (|V | − 1)d}. (In [604] Lin, Miller, Simanjuntak and Slamim called these (a, d)-vertex-antimagic edge labelings). Bodendick and Walther ([185] and [186]) prove the Herschel graph is not (a, d)-antimagic and obtain both positive and negative results about (a, d)-antimagic labelings for various cases of graphs called parachutes P_{g,p}. (P_{g,p} is the graph obtained from the wheel W_{g+p} by deleting p consecutive spokes.) In [89] Bača and Holländer prove that necessary conditions for C_{n} ×P_{2} to be (a, d)-antimagic are d = 1, a = (7n + 4)/2 or d = 3, a = (3n + 6)/2 when n is even, and d = 2, a = (5n + 5)/2 or d = 4, a = (n + 7)/2 when n is odd. Bodendiek and Walther [184] conjectured that C_{n} × P_{2} (n ≥ 3) is ((7n + 4)/2, 1)-antimagic when n is even and is ((5n + 5)/2, 2)- antimagic when n is odd. These conjectures were verified by Bača and Holländer [89] who further proved that C_{n} × P_{2} (n ≥ 3) is ((3n + 6)/2, 3)-antimagic when n is even. Bača and Holländer [89] conjecture that C_{n} × P_{2} is ((n + 7)/2, 4)-antimagic when n is odd and at least 7. Bodendiek and Walther [184] also conjectured that C_{n} × P_{2} (n ≥ 7) is ((n + 7)/2, 4)-antimagic. Miller and Bača [653] prove that the generalized Petersen graph P (n, 2) is ((3n+6)/2, 3)-antimagic for n ≡ 0 (mod 4), n ≥ 8 and conjectured that P (n, k) is ((3n + 6)/2, 3)-antimagic for even n and 2 ≤ k ≤ n/2 − 1 (see §2.7 for the definition of P (n, k).) This conjecture was proved for k = 3 by Xu, Yang, Xi, and Li [975].

Bodendiek and Walther [187] proved that the following graphs are not (a, d)-antimagic: even cycles; paths of even order; stars; C_{3}^{(k)}; C_{4}^{(k)}; trees of odd order at least 5 that have a vertex that is adjacent to three or more end vertices; n-ary trees with at least two layers when d = 1; the Petersen graph; K_{4} and K_{3,3}. They also prove: P_{2k+1} is (k, 1)-antimagic; C_{2k+1} is (k + 2, 1)-antimagic; if a tree of odd order 2k + 1 (k > 1) is (a, d)-antimagic, then d = 1 and a = k; if K_{4k} (k ≥ 2) is (a, d)-antimagic, then d is odd and d ≤ 2k(4k −3)+1; if K_{4k+2} is (a, d)-antimagic, then d is even and d ≤ (2k +1)(4k −1)+1; and if K_{2k+1} (k ≥ 2) is (a, d)-antimagic, then d ≤ (2k + 1)(k − 1). Lin, Miller, Simanjuntak, and Slamin [604] show that no wheel W_{n} (n > 3) has an (a, d)-antimagic labeling.

In [456] Ivančo, and Semaničová show that a 2-regular graph is super edge-magic if and only if it is (a, 1)-antimagic. As a corollary we have that each of the following graphs are (a, 1)-antimagic: kC_{n} for n odd and at least 3; k(C_{3} ∪ C_{n}) for n even and at least 6; k(C_{4} ∪ C_{n}) for n odd and at least 5; k(C_{5} ∪ C_{n}) for n even and at least 4; k(C_{m} ∪ C_{n}) for m even and at least 6, n odd, and n ≥ m/2 + 2. Extending a idea of Kovář they prove if G is (a_{1}, 1)-antimagic and H is obtained from G by adding an arbitrary 2k-factor then H is (a_{2}, 1)-antimagic for some a_{2}. As corollaries they observe that the following graphs are (a, 1)-antimagic: circulant graphs of odd order; 2r-regular Hamiltonian graphs of odd order; and 2r-regular graphs of odd order n < 4r. They further show that if G is an (a, 1)-antimagic r-regular graph of order n and n − r − 1 is a divisor of the non-negative integer a + n(1 + r − (n + 1)/2), then G ⊕ K_{1} is supermagic. As a corollary of this result they have if G is (n − 3)-regular for n odd and n ≥ 7 or (n − 7)-regular for n odd and n ≥ 15, then G ⊕ K_{1} is supermagic.

Yegnanarayanan [991] introduced several variations of antimagic labelings and provides some results about them.

The antiprism on 2n vertices has vertex set {x_{1,1}, …, x_{1,n}, x_{2,1}, …, x_{2,n}} and edge set {x_{j,i}, x_{j,i+1}} ∪ {x_{1,i}, x_{2,i}} ∪ {x_{1,i}, x_{2,i−1}} (subscripts are taken modulo n). For n ≥ 3 and n ≢ 2 (mod 4) Bača [68] gives (6n + 3, 2)-antimagic labelings and (4n + 4, 4)-antimagic labelings for the antiprism on 2n vertices. He conjectures that for n ≡ 2 (mod 4), n ≥ 6, the antiprism on 2n vertices has a (6n+3, 2)-antimagic labeling and a (4n+4, 4)-antimagic labeling.

Nicholas, Somasundaram, and Vilfred [691] prove the following: If K_{m,n} where m ≤ n is (a, d)-antimagic, then d divides ((m − n)(2a + d(m + n − 1)))/4 + dmn/2; if m + n is prime, then K_{m,n}, where n > m > 1, is not (a, d)-antimagic; if K_{n,n+2} is (a, d)-antimagic, then d is even and n + 1 ≤ d < (n + 1)^{2}/2; if K_{n,n+2} is (a, d)-antimagic and n is odd, then a is even and d divides a; if K_{n,n+2} is (a, d)-antimagic and n is even, then d divides 2a; if K_{n,n} is (a, d)-antimagic, then n and d are even and 0 < d < n^{2}/2; if G has order n and is unicylic and (a, d)-antimagic, then (a, d) = (2, 2) when n is even and (a, d) = (2, 2) or (a, d) = ((n + 3)/2, 1) when n is odd; a cycle with m pendant edges attached at each vertex is (a, d)-antimagic if and only if m = 1; the graph obtained by joining an endpoint of P_{m} with one vertex of the cycle C_{n} is (2, 2)-antimagic if m = n or m = n − 1; if m + n is even the graph obtained by joining an endpoint of P_{m} with one vertex of the cycle C_{n} is (a, d)-antimagic if and only if m = n or m = n − 1. They conjecture that for n odd and at least 3, K_{n,n+2} is ((n + 1)(n^{2} − 1)/2, n + 1)-antimagic and they have obtained several results about (a, d)-antimagic labelings of caterpillars.

In [929] Vilfred and Florida proved the following: the one-sided infinite path is (1, 2)- antimagic; P_{2n} is not (a, d)-antimagic for any a and d; P_{2n+1} is (a, d)-antimagic if and only if (a, d) = (n, 1); C_{2n+1} has an (n + 2, 1)-antimagic labeling; and that a 2-regular graph G is (a, d)-antimagic if and only if |V (G)| = 2n + 1 and (a, d) = (n + 2, 1). They also prove that for a graph with an (a, d)-antimagic labeling, q edges, minimum degree δ and maximum degree Δ, the vertex labels lie between δ(δ + 1)/2 and Δ(2q − Δ + 1)/2.

In [930] Vilfred and Florida call a graph G = (V, E) odd antimagic if there exist a bijection f: E → {1, 3, 5, …, 2|E| − 1} such that the induced mapping g_{f}: V → N, defined by g_{f}(v) = ∑{f(uv)| uv ∈ E(G)}, is injective and odd (a, d)-antimagic if there exist positive integers a, d and a bijection f: E → {1, 3, 5, …, 2|E| − 1} such that the induced mapping g_{f}: V → N, defined by g_{f}(v) = ∑{f(uv)| uv ∈ E(G)}, is injective and g_{f}(V) = {a, a + d, a + 2d, …, a + (|V | − 1)d}. Although every (a, d)-antimagic graph is antimagic, C_{4} has an antimagic labeling but does not have an (a, d)-antimagic labeling. They prove: P_{2n+1} is not odd (a, d)-antimagic for any a and d; C_{2n+1} has an odd (2n+2, 2)-antimagic labeling; if a 2-regular graph G has an odd (a, d)-antimagic labeling, then |V (G)| = 2n + 1 and (a, d) = (2n + 2, 2); C_{2n} is odd magic; and an odd magic graph with at least three vertices, minimum degree δ, maximum degree Δ, and q ≥ 2 edges has all its vertex labels between δ^{2} and Δ(2q − Δ).

In Tables 11, 12, and 13 we use the abbreviations A to mean antimagic, (a, d)-A to mean that the graph has an (a, d)-antimagic labeling and (a, d)-EAV to mean that the graph has an (a, d)-antimagic vertex labeling. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The tables were prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.

Table 11: Summary of Antimagic Labelings

Graph  Labeling Notes
P_{n}  A  for n ≥ 3 [405]
C_{n}  A  [405]
W_{n}  A  [405]
K_{n}  A  for n ≥ 3 [405]
every connected graph A?  [405]

except K_{2}

n ≥ 4 vertices A [41]

Δ(G) ≥ n − 2

all complete partite A [41]

graphs except K_{2}

the electronic journal of combinatorics 16 (2009), #DS6 109

Table 12: Summary of (a, d)-Edge-Antimagic Vertex Labelings

Graph  Labeling  Notes
P_{n}  (3, 2)-EAV  [830]
not (a, d)-EAV d > 2 [830]
P_{2n}  (n + 2, 1)-EAV [830]
C_{n}  not (a, d)-EAV d > 1 [98]
C_{2n}  not (a, d)-EAV [830]
C_{2n+1}  (n + 2, 1)-EAV [830]
(n + 3, 1)-EAV [830]
K_{n}  not (a, d)-EAV for n > 1 [98]
K_{n,n}  not (a, d)-EAV for n > 3 [98]
W_{n}  not (a, d)-EAV [98]
C_{3}^{(n)} (friendship graph) (a, 1)-EAV  iff n = 1, 3, 4, 5, 7 [99]
generalized Petersen  not (a, d)-EAV d > 1 [98]

graph P (n, k)

the electronic journal of combinatorics 16 (2009), #DS6 110

Table 13: Summary of (a, d)-Antimagic Labelings

Graph  Labeling  Notes
C_{2n+1}  not (n + 2, 1)-A  n even [187]
P_{2n}  not (a, d)-A  [187]
P_{2n+1}  (n, 1)-A  [187]
stars  not (a, d)-A  [187]
C_{3}^{(k)}, C_{4}^{(k)}  not (a, d)-A  [187]
K_{n,n+2}  (\frac{(n+1)(n^{2}−1)}{2}, n + 1)-A n ≥ 3, n odd [187]
K_{3,3}  not (a, d)-A  [187]
K_{4}  not (a, d)-A  [187]
Petersen graph  not (a, d)-A  [187]
W_{n}  not (a, d)-A  n > 3 [604]
antiprism on 2n  (6n + 3, 2)-A  n ≥ 3, n ≢ 2 (mod 4) [68]
vertices (see §6.1)  (4n + 4, 4)-A  n ≥ 3, n ≢ 2 (mod 4) [68]
(2n + 5, 6)-A?  n ≥ 4 [68]
(6n + 3, 2)-A?  n ≥ 6, n ≢ 2 (mod 4) [68]
(4n + 4, 4)-A?  n ≥ 6, n ≢ 2 (mod 4) [68]
Hershel graph (see [237] not (a, d)-A  [183], [185]
parachutes P_{g,p} (see §6.1) (a, d)-A  for certain classes [183], [185]
C_{n}  not (a, d)-A  n even [187]
prisms C_{n} × P_{2}  ((7n + 4)/2, 1)-A  n ≥ 3, n even [184], [89]
((5n + 5)/2, 2)-A  n ≥ 3, n odd [184], [89]
((3n + 6)/2, 3)-A  n ≥ 3, n even [89]
((n + 7)/2, 4)-A?  n ≥ 7, [185], [89]
generalized Petersen ((3n + 6)/2, 3)-A n ≥ 8, n ≡ 0 (mod 4) [90]

graph P (n, 2)

6.2 (a, d)-Antimagic Total Labelings

Bača, Bertault, MacDougall, Miller, Simanjuntak, and Slamin [83] introduced the notion of a (a, d)-vertex-antimagic total labeling in 2000. For a graph G(V, E), an injective mapping f from V ∪ E to the set {1, 2, …, |V | + |E|} is a (a, d)-vertex-antimagic total labeling if the set {f(v) + ∑ f(vu)} where the sum is over all vertices u adjacent to v for all v in G is {a, a+d, a+2d, …, a+(|V |−1)d}. In the case where the vertex labels are 1,2, …, |V |, (a, d)-vertex-antimagic total labeling is called a super (a, d)-vertex-antimagic total labeling. Among their results are: every super-magic graph has an (a, 1)-vertex-antimagic total labeling; every (a, d)-antimagic graph G(V, E) is (a+|E|+1, d+1)-vertex-antimagic total; and, for d > 1, every (a, d)-antimagic graph G(V, E) is (a + |V | + |E|, d − 1)-vertex-antimagic total. They also show that paths and cycles have (a, d)-vertex-antimagic total labelings for a wide variety of a and d. In [84] Bača et al. use their results in [83] to obtain numerous (a, d)-vertex-antimagic total labelings for prisms, and generalized Petersen graphs (see §2.7 for the definition). (See also [92] and [881] for more results on generalized Petersen graphs.)

Sugeng, Miller, Lin, and Bača [881] prove: C_{n} has a super (a, d)-vertex-antimagic total labeling if and only if d = 0 or 2 and n is odd, or d = 1; P_{n} has a super (a, d)-vertex-antimagic total labeling if and only if d = 2 and n ≥ 3 is odd, or d = 3 and n ≥ 3; no even order tree has a super (a, 1)-vertex antimagic total labeling; no cycle with at least one tail and an even number of vertices has a super (a, 1)-vertex-antimagic labeling; and the star S_{n}, n ≥ 3, has no super (a, d)-super antimagic labeling. As open problems they ask whether K_{n,n} has a super (a, d)-vertex-antimagic total labeling and the generalized Petersen graph has a super (a, d)-vertex-antimagic total labeling for specific values a, d, and n. Lin, Miller, Simanjuntak, and Slamin [604] have shown that for n > 20, W_{n} has no (a, d)-vertex-antimagic total labeling. Tezer and Cahit [911] proved that neither P_{n} nor C_{n} has (a, d)-vertex-antimagic total labelings for a ≥ 3 and d ≥ 6. Kovář [513] has shown that every 2r-regular graph with n vertices has an (s, 1)-vertex antimagic total labeling for s ∈ {(rn + 1)(r + 1) + tn | t = 0, 1, …, r}.

In [684] Ngurah, Baskova, and Simanjuntak provide (a, d)-vertex-antimagic total labelings for the generalized Petersen graphs P (n, m) for the cases: n ≥ 3, 1 ≤ m ≤ ⌊(n − 1)/2⌋, (a, d) = (8n + 3, 2); odd n ≥ 5, m = 2, (a, d) = ((15n + 5)/2, 1); odd n ≥ 5, m = 2, (a, d) = ((21n + 5)/2, 1); odd n ≥ 7, m = 3, (a, d) = ((15n + 5)/2, 1); odd n ≥ 7, m = 3, (a, d) = ((21n + 5)/2, 1); odd n ≥ 9, m = 4, (a, d) = ((15n + 5)/2, 1); and (a, d) = ((21n + 5)/2, 1). They conjecture that for n odd and 1 ≤ m ≤ ⌊(m − 1)/2⌋, P (n, m) has an ((21n + 5)/2, 1)-vertex-antimagic labeling.

Simanjuntak, Bertault, and Miller [830] define an (a,d)-edge-antimagic vertex labeling for a graph G(V, E) as an injective mapping f from V onto the set {1, 2, …, |V |} such that the set {f(u) + f(v)|uv ∈ E} is {a, a + d, a + 2d, …, a + (|E| − 1)d}. (The equivalent notion of (a,d)-indexable labeling was defined by Hegde in 1989 in his Ph. D. thesis–see [414].) Similarly, Simanjuntak et al. define an (a,d)-edge-antimagic total labeling for a graph G(V, E) as an injective mapping f from V ∪ E onto the set {1, 2, …, |V | + |E|} such that the set {f(v) + f(vu) + f(v)|uv ∈ E} where v ranges over all of V is {a, a + d, a + 2d, …, a + (|V | − 1)d}. Among their results are: C_{2n} has no (a, d)-edge-antimagic vertex labeling; C_{2n+1} has a (n + 2, 1)-edge-antimagic vertex labeling and a (n + 3, 1)- edge-antimagic vertex labeling; P_{2n} has a (n+2, 1)-edge-antimagic vertex labeling; P_{n} has a (3, 2)-edge-antimagic vertex labeling; C_{n} has (2n + 2, 1)- and (3n + 2, 1)-edge-antimagic total labelings; C_{2n} has (4n + 2, 2)- and (4n + 3, 2)-edge-antimagic total labelings; C_{2n+1} has (3n + 4, 3)- and (3n + 5, 3)-edge-antimagic total labelings; P_{2n+1} has (3n + 4, 2)-, (3n + 4, 3)-, (2n + 4, 4)-, (5n + 4, 2)-, (3n + 5, 2)-, and (2n + 6, 4)-edge-antimagic total labelings; P_{2n} has (6n, 1)- and (6n+2, 2)-edge-antimagic total labelings; and several parity conditions for (a, d)-edge-antimagic total labelings. They conjecture: C_{2n} has a (2n+3, 4)- or a (2n + 4, 4)-edge-antimagic total labeling; C_{2n+1} has a (n + 4, 5)- or a (n + 5, 5)- edge-antimagic total labeling; paths have no (a, d)-edge-antimagic vertex labelings with d > 2; and cycles have no (a, d)-antimagic total labelings with d > 5. These last two conjectures were verified by Bača, Lin, Miller, and Simanjuntak [98] who proved that a graph with v vertices and e edges that has an (a, d)-edge-antimagic vertex labeling must satisfy d(e−1) ≤ 2v−1−a ≤ 2v−4. As a consequence, they obtain: for every path there is no (a, d)-edge-antimagic vertex labeling with d > 2; for every cycle there is no (a, d)-edge-antimagic vertex labeling with d > 1; for K_{n} (n > 1) there is no (a, d)-edge-antimagic vertex labeling (the cases for n = 2 and n = 3 are handled individually); K_{n,n} (n > 3) has no (a, d)-edge-antimagic vertex labeling; for every wheel there is no (a, d)-edge-antimagic vertex labeling; for every generalized Petersen graph there is no (a, d)-edge-antimagic vertex labeling with d > 1. They also study the relationship between graphs with (a, d)- edge-antimagic labelings and magic and antimagic labelings. They conjecture that every tree has an (a, 1)-edge-antimagic total labeling. Ngurah [682] proved that every odd cycle C_{2k+1} has a (4k + 4, 2)-edge-antimagic total labeling and a (4k + 5, 2)-edge-antimagic total labeling. In [112] Baca and Youssef used parity arguments to find a large number of conditions on p, q and d for which a graph with p vertices and q edges cannot have an (a, d)-edge-antimagic total labeling or vertex-antimagic total labeling. Baca and Youssef [112] made the following connection between (a, d)-edge-antimagic vertex labelings and sequential labelings: if G is a connected graph other than a tree that has an (a, d)-edge-antimagic vertex labeling, then G + K_{1} has a sequential labeling.

In [877] Sudarsana, Ismaimuza, Baskoro, and Assiyatun prove: for every n ≥ 2, P_{n} ∪ P_{n+1} has a (6n + 1, 1)- and a (4n + 3, 3)-edge-antimagic total labeling, for every odd n ≥ 3, P_{n} ∪ P_{n+1} has a (6n, 1)- and a (5n + 1, 2)-edge-antimagic total labeling, for every n ≥ 2, nP_{2} ∪ P_{n} has a (7n, 1)- and a (6n + 1, 2)-edge-antimagic total labeling. In [876] the same authors show that P_{n} ∪ P_{n+1}, nP_{2} ∪ P_{n} (n ≥ 2), and nP_{2} ∪ P_{n+2} are super edgemagic total. They also show that under certain conditions one can construct new super edge-magic total graphs from existing ones by joining a particular vertex of the existing super edge-magic total graph to every vertex in a path or every vertex of a star and by joining one extra vertex to some vertices of the existing graph. Baskoro, Sudarsana, and Cholily [147] also provide algorithms for constructing new super edge-magic total graphs from existing ones by adding pendant vertices to the existing graph. A corollary to one of their results is that the graph obtained by attaching a fixed number of pendant edges to each vertex of a path of even length is super edge-magic. Baskoro and Cholily [145] show that the graphs obtained by attaching any numbers of pendant edges to a single vertex or a fix number of pendant edges to every vertex of the following graphs are super edge-magic total graphs: odd cycles, the generalized Petersen graphs P (n, 2) (n odd and at least 5), and C_{n} × P_{m} (n odd, m ≥ 2).

An (a, d)-edge-antimagic total labeling of G(V, E) is called a super (a,d)-edge-antimagic total if the vertex labels are {1, 2, …, |V (G)|} and the edge labels are {|V (G)|+1, |V (G)|+ 2, …, |V (G)| + |E(G)|}. Bača, Baskoro, Simanjuntak, and Sugeng [82] prove the following: C_{n} has a super (a, d)-edge-antimagic total labeling if and only if either d is 0 or 2 and n is odd, or d = 1; for odd n ≥ 3 and m = 1 or 2, the generalized Petersen graph P (n, m) has a super (11n + 3)/2, 0)-edge-antimagic total labeling and a super ((5n + 5)/2, 2)-edge-antimagic total labeling; for odd n ≥ 3, P (n, (n − 1)/2) has a super ((11n + 3)/2, 0)- edge-antimagic total labeling and a super ((5n + 5)/2, 2)-edge-antimagic total labeling. They also prove: if P (n, m), n ≥ 3, 1 ≤ m ≤ ⌊(n − 1)/2⌋ is super (a, d)-edge-antimagic total, then (a, d) = (4n + 2, 1) if n is even, and either (a, d) = ((11n + 3)/2, 0), or (a, d) = (4n + 2, 1), or (a, d) = ((5n + 5)/2, 2), if n is odd; and for odd n ≥ 3 and m = 1, 2, or (n − 1)/2, P (n, m) has an (a, 0)-edge-antimagic total labeling and an (a, 2)- edge-antimagic total labeling. (In a personal communication MacDougall argues that “edge-magic” is a better term than “(a, 0)-edge-antimagic” for while the latter is technically correct, “antimagic” suggests different weights whereas “magic” emphasizes equal weights and that the edge-magic case is much more important, interesting, and fundamental rather than being just one subcase of equal value to all the others.) They conjecture that for odd n ≥ 9 and 3 ≤ m ≤ (n − 3)/2, P (n, m) has a (a, 0)-edge-antimagic total labeling and an (a, 2)-edge-antimagic total labeling. Ngurah and Baskoro [683] have shown that for odd n ≥ 3, P (n, 1) and P (n, 2) have ((5n+5)/2, 2)-edge-antimagic total labelings and when n ≥ 3 and 1 ≤ m < n/2, P (n, m) has a super (4n + 2, 1)-edge-antimagic total labeling. In [684] Ngurah, Baskova, and Simanjuntak provide (a, d)-edge-antimagic total labelings for the generalized Petersen graphs P (n, m) for the cases m = 1 or 2, odd n ≥ 3, and (a, d) = ((9n + 5)/2, 2).

In [685] Ngurah, Baskoro, and Simanjuntak prove that mC_{n} (n ≥ 3) has an (a, d)- edge-antimagic total in the following cases: (a, d) = (5mn/2 + 2, 1) where m is even; (a, d) = (2mn+2, 2); (a, d) = ((3mn+5)/2, 3) for m and n odd; and (a, d) = ((mn+3), 4) for m and n odd; and mC_{n} has a super (2mn + 2, 1)-edge-antimagic total labeling.

Bača and Barrientos [73] have shown that mK_{n} has a super (a, d)-edge-antimagic total labeling if and only if (i) d ∈ {0, 2}, n ∈ {2, 3} and m ≥ 3 is odd, or (ii) d = 1, n ≥ 2 and m ≥ 2, or (iii) d ∈ {3, 5}, n = 2 and m ≥ 2, or (iv) d = 4, n = 2, and m ≥ 3 is odd. In [72] Bača and Barrientos proved the following: if a graph with q edges and q + 1 vertices has an α-labeling, than it has an (a, 1)-edge-antimagic vertex labeling; a tree has a (3, 2)-edge-antimagic vertex labeling if and only if it has an α-labeling and the number of vertices in its two partite sets differ by at most 1; if a tree with at least two vertices has a super (a, d)-edge-antimagic total labeling, then d is at most 3; if a graph has an (a, 1)-edge-antimagic vertex labeling, then it also has a super (a_{1}, 0)-edge-antimagic total labeling and a super (a_{2}, 2)-edge-antimagic total labeling.

Bača and Youssef [111] proved the following: if G is a connected (a, d)-edge-antimagic vertex graph that is not a tree, then G+K_{1} is sequential; mC_{n} has an (a, d)-edge-antimagic vertex labeling if and only if m and n are odd and d = 1; an odd degree (p, q)-graph G cannot have (a, d)-edge-antimagic total labeling if p ≡ 2 (mod 4) and q ≡ 0 (mod 4), or p ≡ 0 (mod 4), q ≡ 2 (mod 4), and d is even; (p, q)-graph G cannot have a super (a, d)-edge-antimagic total labeling if G has odd degree, p ≡ 2 (mod 4), q is even, and d is odd, or G has even degree, q ≡ 2 (mod 4), and d is even; C_{n} has a (2n + 2, 3)- and an (n + 4, 3)-edge-antimagic total labeling; a (p, q)-graph is not super (a, d)-vertex-antimagic total if: p ≡ 2 (mod 4) and d is even; p ≡ 0 (mod 4), q ≡ 2 (mod 4), and d is odd; p ≡ 0 (mod 8) and q ≡ 2 (mod 4).

In [877] Sudarsana, Ismaimuza, Baskoro, and Assiyatun prove: for every n ≥ 2, P_{n} ∪ P_{n+1} has super (n + 4, 1)- and (2n + 6, 3)-edge antimagic total labelings; for every odd n ≥ 3, P_{n} ∪ P_{n+1} has super (4n + 5, 1)-,(3n + 6, 2)-, (4n + 3, 1)- and (3n + 4, 2)-edge antimagic total labelings; for every n ≥ 2, nP_{2} ∪ P_{n} has super (6n + 2, 1)- and (5n + 3, 2)- edge antimagic total labelings; and for every n ≥ 1, nP_{2} ∪ P_{n+2} has super (6n + 6, 1)- and (5n + 6, 2)-edge antimagic total labelings. They pose a number of open problems about constructing (a, d)-edge antimagic labelings and super (a, d)-edge antimagic labelings for the graphs P_{n} ∪ P_{n+1}, nP_{2} ∪ P_{n}, and nP_{2} ∪ P_{n+2} for specific values of d.

Dafik, Miller, Ryan, and Bača [267] investigated the super edge-antimagicness of the disconnected graph mC_{n} and mP_{n}. For the first case they prove that mC_{n}, m ≥ 2 has a super (a, d)-edge-antimagic total labeling if and only if either d is 0 or 2 and m and n are odd and at least 3, or d = 1, m ≥ 2, and n ≥ 3. For the case of the disjoint union of paths they determine all feasible values for m, n and d for mP_{n} to have a super (a, d)-edge-anti-magic total labeling except when m is even and at least 2, n ≥ 2 and d is 0 or 2. In [269] Dafik, Miller, Ryan, and Bača obtain a number of results about super edge-antimagicness of the disjoint union of two stars and state three open problems.

In [94] Bača, Lascsáková, and Semaničová investigated the connection between graphs with α-labelings and graphs with super (a, d)-edge-antimagic total labelings. Among their results are: If G is a graph with n vertices and n − 1 edges (n ≥ 3) and G has an αlabeling, then mG is super (a, d)-edge-antimagic total if either d is 0 or 2 and m is odd, or d = 1 and n is even; if G has an α-labeling and has n vertices and n − 1 edges with vertex bipartition sets V_{1} and V_{2} where |V_{1}| and |V_{2}| differ by at most 1, then mG is super (a, d)–edge-antimagic total for d = 1 and d = 3. In the same paper Baca et al. prove: caterpillars with odd order at least 3 have super (a, 1)-edge-antimagic total labelings; if G is a caterpillars of odd order at least 3 and G has a super (a, 1)-edge-antimagic total labeling, then mG has a super (b, 1)-edge-antimagic total labeling for some b that is a function of a and m.

In [266] Dafik, Miller, Ryan, and Baca investigated the existence of antimagic labelings of disjoint unions of s-partite graphs. They proved: if s ≡ 0 or 1 (mod 4), s ≥ 4, m ≥ 2, n ≥ 1 or mn is even , m ≥ 2, n ≥ 1, s ≥ 4, then the complete s-partite graph mK_{n,n,...,n} has no super (a, 0)-edge-antimagic total labeling; if m ≥ 2 and n ≥ 1 then mK_{n,n,n,n} has no super (a, 2)-antimagic total labeling; and for m ≥ 2 and n ≥ 1, mK_{n,n,n,n} has an (8mn + 2, 1)-edge-antimagic total labeling. They conjecture that for m ≥ 2, n ≥ 1 and s ≥ 5, the complete s-partite graph mK_{n,n,...,n} has a super (a, 1)-antimagic total labeling.

In [109] Bača, Muntaner-Batle, Semaničová-Feňovčiková, and Shafiq investigate super (a, d)-edge-antimagic total labelings of disconnected graphs. Among their results are: If G is a (super) (a, 2)-edge-antimagic total labeling and m is odd, then mG has a (super) (a′, 2)-edge-antimagic-total labeling where a′ = m(a − 3) + (m + 1)/2 + 2; and if d a positive even integer and k a positive odd integer, G is a graph with all of its vertices having odd degree, and the order and size of G have opposite parity, then 2kG has no (a, d)-edge-antimagic total labeling. Bača and Brankovic [85] have obtained a number of results about the existence of super (a, d)-edge-antimagic totaling of disjoint unions of the form mK_{n,n}. In [86] Bača, Dafik, Miller, and Ryan provide (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings for a variety of disjoint unions of caterpillars. Bača and Youssef [112] proved that mC_{n} has an (a, d)-edge-antimagic vertex labeling if and only if m and n are odd and d = 1. Bača, Dafik, Miller, and Ryan [87] constructed super (a, d)-edge-antimagic total labeling for graphs of the form m(C_{n} ⊙ \overline{K}_{s}) and mP_{n} ∪ kC_{n} while Dafik, Miller, Ryan, and Bača [268] do the same for graphs of the form mK_{n,n,n} and K_{1,m} ∪ 2sK_{1,n}. Both papers provide a number of open problems. In [101] Bača, Lin, and Muntaner-Batle provide super (a, d)-edge-antimagic total labeling of forests in which every component is a specific kind of tree.

In [99] Bača, Lin, Miller, and Youssef prove: if the friendship C_{3}^{(n)} is super (a, d)- antimagic total, then d < 3; C_{3}^{(n)} has an (a, 1)-edge antimagic vertex labeling if and only if n = 1, 3, 4, 5 and 7; C_{3}^{(n)} has a super (a, d)-edge-antimagic total labelings for d = 0 and 2; C_{3}^{(n)} has a super (a, 1)-edge-antimagic total labeling; if a fan F_{n} (n ≥ 2) has a super (a, d)-edge-antimagic total labeling, then d < 3; F_{n} has a super (a, d)-edge-antimagic total labeling if 2 ≤ n ≤ 6 and d = 0, 1 or 2; the wheel W_{n} has a super (a, d)-edge-antimagic total labeling if and only if d = 1 and n ≢ 1 (mod 4); K_{n}, n ≥ 3, has a super (a, d)- edge-antimagic total labeling if and only if either d = 0 and n = 3, or d = 1 and n ≥ 3, or d = 2 and n = 3; K_{n,n} has a super (a, d)-edge antimagic total labeling if and only if d = 1 and n ≥ 2.

Bača, Lin, and Muntaner-Batle [100] have shown that if a tree with at least two vertices has a super (a, d)-edge-antimagic total labeling, then d is at most three and P_{n}, n ≥ 2, has a super (a, d)-edge-antimagic total labeling if and only if d = 0, 1, 2, or 3. They also characterize certain path-like graphs in a grid that have super(a, d)-edge-antimagic total labelings.

Recall that C_{n}^{t} denotes the graph obtained from the n-cycle by joining two vertices at a distance t. MacDougall and Wallis [638] have proved the following: C_{4m+3}^{t}, m ≥ 1, has a super (a, 0)-edge-antimagic total labeling for all possible values of t with a = 10m+9 or 10m + 10; C_{4m+1}^{t}, m ≥ 3, has a super (a, 0)-edge-antimagic total labeling for all possible values except t = 5, 9, 4m − 4, and 4m − 8 with a = 10m + 4 and 10m + 5; C_{4m+1}^{t}, m ≥ 1, has a super (10m+4, 0)-edge-antimagic total labeling for all t ≡ 1 (mod 4) except 4m−3; C_{4m}^{t}, m > 1, has a super (10m + 2, 0)-edge-antimagic total labeling for all t ≡ 2 (mod 4); C_{4m+2}^{t}, m > 1, has a super (10m + 7, 0)-edge-antimagic total labeling for all odd t other than 5 and for t = 2 or 6.

Bača and Murugan [110] have proved: if C_{n}^{t}, n ≥ 4, 2 ≤ t ≤ n − 2, is super (a, d)- edge-antimagic total, then d = 0, 1, or 2; for n = 2k + 1 ≥ 5, C_{n}^{t} has a super (a, 0)- edge-antimagic total labeling for all possible values of t with a = 5k + 4 or 5k + 5; for n = 2k+1 ≥ 5, C_{n}^{t} has a super (a, 2)-edge-antimagic total labeling for all possible values of t with a = 3k +3 or 3k +4; for n ≡ 0 (mod 4), C_{n}^{t} has a super (5n/2+2, 0)-edge-antimagic total labeling and a super (3n/2+2, 0)-edge-antimagic total labeling for all t ≡ 2 (mod 4); for n = 10 and n ≡ 2 (mod 4), n ≥ 18, C_{n}^{t} has a super (5n/2 + 2, 0)-edge-antimagic total labeling and a super (3n/2 + 2, 0)-edge-antimagic total labeling for all t ≡ 3 (mod 4) and for t = 2 and 6; for odd n ≥ 5, C_{n}^{t} has a super (2n + 2, 1)-edge-antimagic total labeling for all possible values of t; for even n ≥ 6, C_{n}^{t} has a super (2n + 2, 1)-edge-antimagic total labeling for all odd t ≥ 3; for even n ≡ 0 (mod 4), n ≥ 4, C_{n}^{t} has a super (2n + 2, 1)- edge-antimagic total labeling for all t ≡ 2 (mod 4). They conjecture that there is a super (2n + 2, 1)-edge-antimagic total labeling of C_{n}^{t} for n ≡ 0 (mod 4) and for t ≡ 0 (mod 4) and for n ≡ 2 (mod 4) and for t even.

In [880] Sugeng, Miller, and Bača prove that the ladder, P_{n} × P_{2}, is super (a, d)-edge-antimagic total if n is odd and d = 0, 1, or 2 and P_{n}×P_{2} is super (a, 1)-antimagic total if n is even. They conjecture that P_{n} × P_{2} is super (a, 0)- and (a, 2)-edge-antimagic when n is even. Sugeng, Miller, and Bača [880] prove that C_{m} ×P_{2} has a super (a, d)-edge-antimagic total labeling if and only if either d = 0, 1 or 2 and m is odd and at least 3, or d = 1 and m is even and at least 4. They conjecture that if m is even, m ≥ 4, n ≥ 3, and d = 0 or 2, then C_{m} × P_{n} has a super (a, d)-edge-antimagic total labeling.

Sugeng, Miller, and Bača [880] define a variation of a ladder, 𝕃_{n}, as the graph obtained from P_{n} × P_{2} by joining each vertex u_{i} of one path to the vertex v_{i+1} of the other path for i = 1, 2, …, n − 1. They prove 𝕃_{n}, n ≥ 2, has a super (a, d)-edge-antimagic total labeling if and only if d = 0, 1, or 2.

In [74] Bača, Bashir and Semaničová showed that for n ≥ 4 and d = 0, 1, 2, 3, 4, 5, and 6 the antiprism A_{n} has a super d-antimagic labeling of type (1, 1, 1). The generalized antiprism A_{m}^{n} is obtained from C_{m}×P_{n} by inserting the edges {v_{i,j+1}, v_{i+1,j}} for 1 ≤ i ≤ m and 1 ≤ j ≤ n − 1 where the subscripts are taken modulo m. Sugeng et al. prove that A_{m}^{n}, m ≥ 3, n ≥ 2, is super (a, d)-edge-antimagic total if and only if d = 1.

Sugeng, Miller, Slamin, and Bača [883] proved: the star S_{n} has a super (a, d)-antimagic total labeling if and only if either d = 0, 1 or 2, or d = 3 and n = 1 or 2; if a nontrivial caterpillar has a super (a, d)-edge-antimagic total labeling, then d ≤ 3; all caterpillars have super (a, 0)-, (a, 1)- and (a, 2)-edge-antimagic total labelings; all caterpillars have a super (a, 1)-edge-antimagic total labeling; if m and n differ by at least 2 the double star S_{m,n} (that is, the graph obtained by joining the centers of K_{1,m} and K_{1,n} with an edge) has no (a, 3)-edge-antimagic total labeling.

Sugeng and Miller [878] show how to manipulate adjacency matrices of graphs with (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings to obtain new (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings. Among their results are: every graph can be embedded in a connected (a, d)- edge-antimagic vertex graph; every (a, d)-edge-antimagic vertex graph has a proper (a, d)- edge-antimagic vertex subgraph; if a graph has a (a, 1)-edge-antimagic vertex labeling and an odd number of edges, then it has a super (a, 1)-edge-antimagic total labeling; every super edge magic total graph has an (a, 1)-edge-antimagic vertex labeling; and every graph can be embedded in a connected super (a, d)-edge-antimagic total graph.

Bača, Lin, Miller, and Ryan [97] define a Möbius grid, M_{n}^{m}, as the graph with vertex set {x_{i,j} for i = 1, 2, …, m + 1, j = 1, 2, …, n} and edge set {x_{i,j}x_{i,j+1}| i = 1, 2, …, m + 1, j = 1, 2, …, n − 1} ∪ {x_{i,j}x_{i+1,j}| i = 1, 2 …, m, j = 1, 2, …, n} ∪ {x_{i,n}x_{m+2−i,1}| i = 1, 2, …, m + 1}. They prove that for n ≥ 2 and m ≥ 4, M_{n}^{m} has no d-antimagic vertex labeling with d ≥ 5 and no d-antimagic-edge labeling with d ≥ 9.

The book [106] by Bača and Miller has a wealth of material and open problems on super edge-antimagic labelings. In [81] Bača, Baskoro, Miller, Ryan, Simanjuntak, and Sugeng provide detailed survey of results on edge antimagic labelings and include many conjectures and open problems.

In Tables 14 and 15 we use the abbreviations

(a, d)-VAT (a, d)-vertex-antimagic total labeling

(a, d)-SVAT super (a, d)-vertex-antimagic total labeling

(a, d)-EAT (a, d)-edge-antimagic total labeling

(a, d)-SEAT super (a, d)-edge-antimagic total labeling

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.

Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)-Vertex-Antimagic Total Labelings

Graph  Labeling  Notes
P_{n}  (a, d)-VAT  wide variety of a and d [83]
P_{n}  (a, d)-SVAT  iff d = 3, d = 2, n ≥ 3 odd
or d = 3, n ≥ 3 [881]
C_{n}  (a, d)-VAT  wide variety of a and d [82]
C_{n}  (a, d)-SVAT  iff d = 0, 2 and n odd or d = 1 [881]
generalized Petersen (a, d)-VAT  [84]
graph P (n, k)  (a, 1)-VAT  n ≥ 3, 1 ≤ k ≤ n/2 [882]
prisms C_{n} × P_{2}  (a, d)-VAT  [84]
antiprisms  (a, d)-VAT  [84]
W_{n}  not (a, d)-VAT for n > 20 [604]
K_{1,n}  not (a, d)-SVAT n ≥ 3 [881]

Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings

Graph Labeling  Notes
P_{n}  not (a, d)-EAT d > 2 [98]
P_{2n}  (6n, 1)-EAT  [830]
(6n + 2, 2)-EAT [830]
P_{2n+1} (3n + 4, 2)-EAT [830]
(3n + 4, 3)-EAT [830]
(2n + 4, 4)-EAT [830]
(5n + 4, 2)-EAT [830]
(3n + 5, 2)-EAT [830]
(2n + 6, 4)-EAT [830]
C_{n}  (2n + 2, 1)-EAT [830]
(3n + 2, 1)-EAT [830]
not (a, d)-EAT d > 5 [98]
C_{2n}  (4n + 2, 2)-EAT [830]
(4n + 3, 2)-EAT [830]
(2n + 3, 4)-EAT? [830]
(2n + 4, 4)-EAT? [830]
C_{2n+1} (3n + 4, 3)-EAT [830]
(3n + 5, 3)-EAT [830]
(n + 4, 5)-EAT? [830]
(n + 5, 5)-EAT? [830]

the electronic journal of combinatorics 16 (2009), #DS6 120

Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings continued

Graph  Labeling  Notes
trees  (a, 1)-EAT?  [98]
K_{n}  not (a, d)-EAT  d > 5 [98]
K_{n,n}  (a, d)-EAT  iff d = 1, n ≥ 2 [99]
caterpillars  (a, d)-EAT  d ≤ 3 [883]
W_{n}  not (a, d)-EAT  d > 4 [98]
generalized Petersen not (a, d)-EAT  d > 4 [98]
graph P (n, k)  ((5n + 5)/2, 2)-EAT for n odd, n ≥ 3 and k = 1, 2 [683]
super (4n + 2, 1)-EAT for n ≥ 3, and 1 ≤ k ≤ n/2 [683]

Table 16: Summary of (a, d)-Super-Edge-Antimagic Total Labelings

Graph  Labeling  Notes
C_{n}^{+} (see §2.2)  (a, d)-SEAT variety of cases [65], [110]
P_{n} × P_{2} (ladders)  (a, d)-SEAT n odd, d ≤ 2 [880]
n even, d = 1 [880]
(a, d)-SEAT? d = 0, 2, n even [880]
C_{n} × P_{2}  (a, d)-SEAT iff d ≤ 3 n odd [880]
or d = 1, n ≥ 4 even [880]
C_{m} × P_{n}  (a, d)-SEAT? m ≥ 4 even, n ≥ 3, d = 0, 2 [880]
caterpillars  (a, 1)-SEAT [883]
C_{3}^{(n)} (friendship graphs) (a, d)-SEAT d = 0, 1, 2 [99]
F_{n} (n ≥ 2) (fans)  (a, d) SEAT only if d < 3 [99]
(a, d)-SEAT 2 ≤ n ≤ 6, d = 0, 1, 2 [99]
W_{n}  (a, d)-SEAT iff d = 1, n ≢ 1 (mod 4) [99]
K_{n} (n ≥ 3)  (a, d) SEAT iff d = 0, n = 3 [99]
d = 1, n ≥ 3 [99]
d = 2, n = 3 [99]
trees  (a, d)-SEAT only if d ≤ 3 [100]
P_{n} (n > 1)  (a, d)-SEAT iff d ≤ 3 [100]
mK_{n}  (a, d)-SEAT iff d ∈ {0, 2}, n ∈ {2, 3}, m ≥ 3 odd [73]
d = 1, m, n ≥ 2 [73]
d = 3 or 5,n = 2, m ≥ 2 [73]
d = 4, n = 2, m ≥ 3 odd [73]
C_{n}  (a, d)-SEAT iff d = 0 or 2, n odd [100]
d = 1 [82]
P (m, n)  (a, d)-SEAT many cases [82]

6.3 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1)

Bača [67] defines a connected plane graph G with edge set E and face set F to be (a, d)- face antimagic if there exist positive integers a and d and a bijection g: E → {1, 2, …, |E|} such that the induced mapping ψ_{g}: F → {a, a + d, …, a + (|F (G)| − 1)d}, where for a face f, ψ_{g}(f) is the sum of all g(e) for all edges e surrounding f is also a bijection. Bača [67] and Bača and Miller [103] describe (a, d)-face antimagic labelings for a certain classes of convex polytopes.

In [69] Bača proves that for n even and at least 4, the prism C_{n} × P_{2} is (6n + 3, 2)-face antimagic and (4n + 4, 4)-face antimagic. He also conjectures that C_{n} × P_{2} is (2n + 5, 6)- face antimagic. In [95] Bača, Lin, and Miller investigate (a, d)-face antimagic labelings of the convex polytopes P_{m+1} × C_{n}. They show that if these graphs are (a, d)-face antimagic then either d = 2 and a = 3n(m + 1) + 3, or d = 4 and a = 2n(m + 1) + 4, or d = 6 and a = n(m + 1) + 5. They also prove that if n is even, n ≥ 4 and m ≡ 1 (mod 4), m ≥ 3, then P_{m+1} × C_{n} has a (3n(m + 1) + 3, 2)-face antimagic labeling and if n is at least 4 and even and m is at least 3 and odd, or if n ≡ 2 (mod 4), n ≥ 6 and m is even, m ≥ 4, then P_{m+1} × C_{n} has a (3n(m + 1) + 3, 2)-face antimagic labeling and a (2n(m + 1) + 4, 4)- face antimagic labeling. They conjecture that P_{m+1} × C_{n} has (3n(m + 1) + 3, 2)- and (2n(m + 1) + 4, 4)-face antimagic labelings when m ≡ 0 (mod 4), n ≥ 4, and for m even and m ≥ 4, that P_{m+1} × C_{n} has a (n(m + 1) + 5, 6)-face antimagic labeling when n is even and at least 4.

In [104] Bača and Miller define the class Q_{n}^{m} of convex polytopes with vertex set {y_{j,i} : i = 1, 2, …, n; j = 1, 2, …, m+1} and edge set {y_{j,i}y_{j,i+1} : i = 1, 2, …, n; j = 1, 2, …, m+ 1} ∪ {y_{j,i}y_{j+1,i} : i = 1, 2, …, n; j = 1, 2, …, m} ∪ {y_{j,i+1}y_{j+1,i} : 1 + 1, 2, …, n; j = 1, 2, …, m, j odd} ∪ {y_{j,i}y_{j+1,i+1} : i = 1, 2, …, n; j = 1, 2, …, m, j even} where y_{j,n+1} = y_{j,1}. They prove that for m odd, m ≥ 3, n ≥ 3, Q_{n}^{m} is (7n(m + 1)/2 + 2, 1)-face antimagic and when m and n are even, m ≥ 4, n ≥ 4, Q_{n}^{m} is (7n(m + 1)/2 + 2, 1)-face antimagic. They conjecture that when n is odd, n ≥ 3, and m is even, then Q_{n}^{m} is ((5n(m + 1) + 5)/2, 2)−face antimagic and ((n(m + 1) + 7)/2, 4)-face antimagic. They further conjecture that when n is even, n > 4, m > 1 or n is odd, n > 3 and m is odd, m > 1, then Q_{n}^{m} is (3n(m + 1)/2 + 3, 3)-face antimagic. In [71] Bača proves that for the case m = 1 and n ≥ 3 the only possibilities for (a, d)-antimagic labelings for Q_{n}^{m} are (7n + 2, 1) and (3n + 3, 3). He provides the labelings for the first case and conjectures that they exist for the second case.

In [77] Bača et al. provide a detailed survey of results on face antimagic labelings and include many conjectures and open problems.

For a plane graph G, Bača and Miller [105] call a bijection h from V (G)∪E(G)∪F (G) to {1, 2, …, |V (G)| + |E(G)| ∪ |F (G)|} a d-antimagic labeling of type (1, 1, 1) if for every number s the set of s-sided face weights is W_{s} = {a_{s}, a_{s} + d, a_{s} + 2d, …, a_{s} + (f_{s} − 1)d} for some integers a_{s} and d, where f_{s} is the number of s-sided faces (W_{s} varies with s). They show that the prisms C_{n} × P_{2} (n ≥ 3) have a 1-antimagic labeling of type (1, 1, 1) and that for n ≡ 3 (mod 4), C_{n} ×P_{2} have a d-antimagic labeling of type (1, 1, 1) for d = 2, 3, 4, and 6. They conjecture that for all n ≥ 3, C_{n} × P_{2} has a d-antimagic labeling of type (1, 1, 1) for d = 2, 3, 4, 5, and 6. This conjecture has been proved for the case d = 3 and n ≠ 4 by Bača, Miller, and Ryan [107] (the case d = 3 and n = 4 is open). The cases for d = 2, 4, 5, and 6 were done by Lin, Slamin, Bača, and Miller [605]. Bača, Lin, and Miller [96] prove: for m, n > 8, P_{m} × P_{n} has no d-antimagic edge labeling of type (1, 1, 1) with d ≥ 9; for m ≥ 2, n ≥ 2, and (m, n) ≠ (2, 2), P_{m} × P_{n} has d-antimagic labelings of type (1, 1, 1) for d = 1, 2, 3, 4, and 6. They conjecture the same is true for d = 5.

Bača, Miller, and Ryan [107] also prove that for n ≥ 4 the antiprism (see §6.1 for the definition) on 2n vertices has a d-antimagic labeling of type (1, 1, 1) for d = 1, 2, and 4. They conjecture the result holds for d = 3, 5, and 6 as well. Lin, Ahmad, Miller, Sugeng and Bača [602] did the cases that d = 7 for n ≥ 3 and d = 12 for n ≥ 11. Sugeng, Miller, Lin, and Bača [882] did the cases: d = 7, 8, 9, 10 for n ≥ 5; d = 15 for n ≥ 6; d = 18 for n ≥ 7; d = 12, 14, 17, 20, 21, 24, 27, 30, 36 for n odd and n ≥ 7; d = 16, 26 for n odd and n ≥ 9.

Bača, Jendraľ, Miller, and Ryan [92] prove: for n even, n ≥ 6, the generalized Petersen graph P (n, 2) has a 1-antimagic labeling of type (1, 1, 1); for n even, n ≥ 6, n ≠ 10, and d = 2 or 3, P (n, 2) has a d-antimagic labeling of type (1, 1, 1); and for n ≡ 0 (mod 4), n ≥ 8 and d = 6 or 9, P (n, 2) has a d-antimagic labeling of type (1, 1, 1). They conjecture that there is an d-antimagic labeling of type (1,1,1) for P (n, 2) when n ≡ 2 (mod 4), n ≥ 6, and d = 6 or 9.

Bača, Baskoro, and Miller [79] have proved that hexagonal planar honeycomb graphs with an even number of columns have 2-antimagic and 4-antimagic labelings of type (1, 1, 1). They conjecture that these honeycombs also have d-antimagic labelings of type (1, 1, 1) for d = 3 and 5. They pose the odd number of columns case for 1 ≤ d ≤ 5 as an open problem. Bača, Baskoro, and Miller [80] give d-antimagic labelings of a special class of plane graphs with 3-sided internal faces for d = 0, 2, and 4. Bača, Lin, Miller, and Ryan [97] prove for odd n ≥ 3, m ≥ 1 and d = 0, 1, 2 or 4, the Möbius grid M_{n}^{m} has an d-antimagic labeling of type (1, 1, 1).

Kathiresan and Ganesan [481] define a class of plane graphs denoted by P_{a}^{b} (a ≥ 3, b ≥ 2) as the graph obtained by starting with vertices v_{1}, v_{2}, …, v_{a} and for each i = 1, 2 …, a − 1 joining v_{i} and v_{i+1} with b internally disjoint paths of length i + 1. They prove that P_{a}^{b} has d-antimagic labelings of type (1, 1, 1) for d = 0, 1, 2, 3, 4 and 6. Lin and Sugen [606] prove that P_{a}^{b} has a d-antimagic labeling of type (1, 1, 1) for d = 5, 7a − 2, a + 1, a − 3, a − 7, a + 5, a − 4, a + 2, 2a − 3, 2a − 1, a − 1, 3a − 3, a + 3, 2a + 1, 2a + 3, 3a + 1, 4a − 1, 4a − 3, 5a − 3, 3a − 1, 6a − 5, 6a − 7, 7a − 7, and 5a − 5. Similarly, Bača, Baskoro, and Cholily [76] define a class of plane graphs denoted by C_{a}^{b} as the graph obtained by starting with vertices v_{1}, v_{2}, …, v_{a} and for each i = 1, 2 …, a joining v_{i} and v_{i+1} with b internally disjoint paths of length i + 1 (subscripts are taken modulo a). In [76] and [75] they prove that for a ≥ 3 and b ≥ 2, C_{a}^{b} has a d-antimagic labeling of type (1, 1, 1) for d = 0, 1, 2, 3, a + 1, a − 1, a + 2, and a − 2.

In the table following we use the abbreviations (a, d)-FA (a, d)-face antimagic labeling d-AT(1,1,1) d-antimagic labeling of type (1, 1, 1).

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.

Table 17: Summary of Face Antimagic Labelings

Graph  Labeling  Notes
Q_{n}^{m} (see §6.3) (7n(m + 1)/2 + 2, 1)-FA  m ≥ 3, n ≥ 3, m odd [104]
(7n(m + 1)/2 + 2, 1)-FA  m ≥ 4, n ≥ 4, m, n even [104]
((5n(m + 1) + 5)/2, 2)-FA? m ≥ 2, n ≥ 3, m even, n odd [104]
((n(m + 1) + 7)/2, 4)-FA? m ≥ 2, n ≥ 3, m even, n odd [104]
(3n(m + 1)/2 + 3, 3)-FA? m > 1, n > 4, n even [104]
(3n(m + 1)/2 + 3, 3)-FA? m > 1, n > 3, m odd, n odd [104]
C_{n} × P_{2}  (6n + 3, 2)-FA  n ≥ 4, n even [69]
(4n + 4, 4)-FA  n ≥ 4, n even [69]
(2n + 5, 6)-FA?  [69]
P_{m+1} × C_{n}  (3n(m + 1) + 3, 2)-FA  n ≥ 4, n even and [95]

m ≥ 3, m ≡ 1 (mod 4),

(3n(m + 1) + 3, 2)-FA and n ≥ 4, n even and [95]
(2n(m + 1) + 4, 4)-FA  m ≥ 3, m odd [95],

or n ≥ 6, n ≡ 2 (mod 4) and m ≥ 4, m even

(3n(m + 1) + 3, 2)-FA?  m ≥ 4, n ≥ 4, m ≡ 0 (mod 4) [95]
(2n(m + 1) + 4, 4)-FA?  m ≥ 4, n ≥ 4, m ≡ 0 (mod 4) [95]
(n(m + 1) + 5, 6)-FA?  n ≥ 4, n even [95]

Table 18: Summary of d-antimagic Labelings of Type (1,1,1)

Graph  Labeling  Notes
P_{m} × P_{n}  not d-AT(1,1,1) m, n, d ≥ 9, [96]
P_{m} × P_{n}  d-AT(1,1,1)  d = 1, 2, 3, 4, 6; m, n ≥ 2, (m, n) ≠ (2, 2) [96]
P_{m} × P_{n}  5-AT(1,1,1)  m, n ≥ 2, (m, n) ≠ (2, 2) [96]
C_{n} × P_{2}  1-AT(1,1,1)  [105]
d-AT(1,1,1)  d = 2, 3, 4 and 6 [105]
for n ≡ 3 (mod 4)
d-AT(1,1,1)  d = 2, 4, 5, 6 for n ≥ 3 [605]
d-AT(1,1,1)  d = 3 for n ≥ 5 [107]
P_{m} × P_{n}  5-AT(1,1,1)?  [605]
not d-AT  m, n > 8, d ≥ 9 [605]
antiprism on 2n  d-AT(1,1,1)  d = 1, 2 and 4 for n ≥ 4 [107]
vertices  d-AT(1,1,1)?  d = 3, 5 and 6 for n ≥ 4 [107]
M_{n}^{m} (Möbius grids)  d-AT(1,1,1)  n ≥ 3 odd, d = 0, 1, 2, 4 [97]
d = 7, n ≥ 3 [602]
d = 12, n ≥ 11 [602]
d = 7, 8, 9, 10, n ≥ 5 [882]
d = 15, n ≥ 6 [882]
d = 18 n ≥ 7 [882]
P (n, 2)  d-AT(1,1,1)  d = 1; d = 2, 3, n ≥ 6, n ≠ 10 [92]
P (4n, 2)  d-AT(1,1,1)  d = 6, 9, n ≥ 2, n ≠ 10 [92]
P (4n + 2, 2)  d-AT(1,1,1)?  d = 6, 9, n ≥ 1, n ≠ 10 [92]
honeycomb graphs with d-AT(1,1,1)  d = 2, 4 [79]
even number of columns d-AT(1,1,1)?  d = 3, 5 [79]
C_{n} × P_{2}  d-AT(1,1,1)  d = 1, 2, 4, 5, 6 [605], [105]
C_{n} × P_{2}  3-AT(1,1,1)  n ≠ 4 [107]

6.4 Product Antimagic Labelings

Figueroa-Centeno, Ichishima, and Muntaner-Batle [321] have introduced multiplicative analogs of magic and antimagic labelings. They define a graph G of size q to be product magic if there is a labeling from E(G) onto {1, 2, …, q} such that, at each vertex v, the product of the labels on the edges incident with v is the same. They call a graph G of size q product antimagic if there is a labeling f from E(G) onto {1, 2, …, q} such that the products of the labels on the edges incident at each vertex v are distinct. They prove: a graph of size q is product magic if and only if q ≤ 1 (that is, if and only if it is K_{2}, \overline{K_{n}} or K_{2} ∪ \overline{K_{n}}); P_{n} (n ≥ 4) is product antimagic; every 2-regular graph is product antimagic; and, if G is product antimagic, then so are G + K_{1} and G ⊙ \overline{K}_{n}. They conjecture that a connected graph of size q is product antimagic if and only if q ≥ 3. Kaplan, Lev, and Roditty [474] proved the following graphs are product anti-magic: the disjoint union of cycles and paths where each path has least three edges; connected graphs with n vertices and m edges where m ≥ 4nln n; graphs G = (V, E) where each component has at least two edges and δ(G) ≥ 8\sqrt{ln |E| ln (ln |E|)}; all complete k-partite graphs except K_{2} and K_{1,2}; and G ⊙ H where G has no isolated vertices and H is regular

In [707] Pikhurko characterizes all large graphs that are product anti-magic graphs. More precisely, it is shown that there is an n_{0} such that a graph with n ≥ n_{0} vertices is product anti-magic if and only if it does not belong to any of the following four classes: graphs that have at least one isolated edge; graphs that have at least two isolated vertices; unions of vertex-disjoint of copies of K_{1,2}; and graphs consisting of one isolated vertex and vertex-disjoint spiders where the vertex-disjoint spider S_{k,t} is a graph obtained by subdividing some t edges of the star K_{1,k+l}.

In [321] Figueroa-Centeno, Ichishima, and Muntaner-Batle also define a graph G with p vertices and q edges to be product edge-magic if there is a labeling f from V (G) ∪ E(G) onto {1, 2, …, p+q} such that f(u)·f(v)·f(uv) is a constant for all edges uv and product edge-antimagic if there is a labeling f from V (G) ∪ E(G) onto {1, 2, …, p + q} such that for all edges uv the products f(u)·f(v)·f(uv) are distinct. They prove K_{2}∪\overline{K}_{n} is product edge-magic, a graph of size q without isolated vertices is product edge-magic if and only if q ≤ 1 and every graph other than K_{2} and K_{2} ∪ \overline{K}_{n} is product edge-antimagic.

7 Miscellaneous Labelings

7.1 Sum Graphs

In 1990, Harary [399] introduced the notion of a sum graph. A graph G(V, E) is called a sum graph if there is an bijection f from V to a set of positive integers S such that xy ∈ E if and only if f(x) + f(y) ∈ S. Since the vertex with the highest label in a sum graph cannot be adjacent to any other vertex, every sum graph must contain isolated vertices. In 1991 Harary, Hentzel, and Jacobs [401] defined a real sum graph in an analogous way by allowing S to be any finite set of positive real numbers. However, they proved that every real sum graph is a sum graph. Bergstrand, Hodges, Jennings, Kuklinski, Wiener, and Harary [157] defined a product graph analogous to a sum graph except that 1 is not permitted to belong to S. They proved that every product graph is a sum graph and vice versa.

For a connected graph G, let σ(G), the sum number of G, denote the minimum number of isolated vertices that must be added to G so that the resulting graph is a sum graph (some authors use s(G) for the sum number of G). A labeling that makes G together with σ(G) isolated points a sum graph is called an optimal sum graph labeling. Ellingham [296] proved the conjecture of Harary [399] that σ(T) = 1 for every tree T ≠ K_{1}. Smyth [857] proved that there is no graph G with e edges and σ(G) = 1 when n^{2}/4 < e ≤ n(n − 1)/2. Smyth [858] conjectures that the disjoint union of graphs with sum number 1 has sum number 1. More generally, Kratochvil, Miller, and Nguyen [515] conjecture that σ(G ∪ H) ≤ σ(G) + σ(H) − 1. Hao [398] has shown that if d_{1} ≤ d_{2} ≤ ⋯ ≤ d_{n} is the degree sequence of a graph G, then σ(G) > max(d_{i} − i) where the maximum is taken over all i. Bergstand et al. [156] proved that σ(K_{n}) = 2n − 3. Hartsfield and Smyth [406] claimed to have proved that σ(K_{m,n}) = ⌈3m + n − 3⌉/2 when n ≥ m but Yan and Liu [977] found counterexamples to this assertion when m ≠ n. Pyatkin [712], Liaw, Kuo, and Chang [599], Wang and Liu [948], and He, Shen, Wang, Chang, Kang, and Yu [410] have shown that for 2 ≤ m ≤ n, σ(K_{m,n}) = ⌈\frac{n}{p} + \frac{(p+1)(m−1)}{2}⌉ where p = ⌈\sqrt{\frac{2n}{m−1} + \frac{1}{4}} − \frac{1}{2}⌉

is the unique integer such that \frac{(p−1)p(m−1)}{2} < n ≤ \frac{(p+1)p(m−1)}{2}.

Miller, Ryan, Slamin, and Smyth [659] proved that σ(W_{n}) = \frac{n}{2} + 2 for n even and σ(W_{n}) = n for n ≥ 5 and n odd (see also [898]). Miller, Ryan, and Smyth [661] prove that the complete n-partite graph on n sets of 2 nonadjacent vertices has sum number 4n − 5 and obtain upper and lower bounds on the complete n-partite graph on n sets of m nonadjacent vertices. Fernau, Ryan, and Sugeng [319] proved that the generalized friendship graphs C_{n}^{(t)} (see §2.2) has sum number 2 except for C_{4}. Gould and Rödl [375] investigated bounds on the number of isolated points in a sum graph. A group of six undergraduate students [374] proved that σ(K_{n} − edge) ≤ 2n − 4. The same group of six students also investigated the difference between the largest and smallest labels in a sum graph, which they called the spum. They proved spum of K_{n} is 4n − 6 and the spum of C_{n} is at most 4n − 10. Kratochvil, Miller, and Nguyen [515] have proved that every sum graph on n vertices has a sum labeling such that every label is at most 4^{n}.

At a conference in 2000 Miller [652] posed the following two problems: Given any graph G, does there exist an optimal sum graph labeling that uses the label 1; Find a class of graphs G that have sum number of the order |V (G)|^{s} for s > 1. (Such graphs were shown to exist for s = 2 by Gould and Rödl in [375]).

In [846] Slamet, Sugeng, and Miller show how one can use sum graph labelings to distribute secret information to set of people so that only authorized subsets can reconstruct the secret.

Chang [229] generalized the notion of sum graph by permitting x = y in the definition of sum graph. He calls graphs that have this kind of labeling strong sum graphs and uses i^{∗}(G) to denote the minimum positive integer m such that G∪mK_{1} is a strong sum graph. Chang proves that i^{∗}(K_{n}) = σ(K_{n}) for n = 2, 3, and 4 and i^{∗}(K_{n}) > σ(K_{n}) for n ≥ 5. He further shows that for n ≥ 5, 3n^{log_{2}3} > i^{∗}(K_{n}) ≥ 12⌊n/5⌋ − 3.

In 1994 Harary [400] generalized sum graphs by permitting S to be any set of integers. He calls these graphs integral sum graphs. Unlike sum graphs, integral sum graphs need not have isolated vertices. Sharary [801] has shown that C_{n} and W_{n} are integral sum graphs for all n ≠ 4. Chen [244] proved that trees obtained from a star by extending each edge to a path and trees all of whose vertices of degree not 2 are at least distance 4 apart are integral sum graphs. He conjectures that all trees are integral sum graphs. In [244] and [246] Chen gives methods for constructing new connected integral sum graphs from given integral sum graphs by identifying vertices. Chen [246] has shown that every graph is an induced subgraph of a connected integral sum graph. Chen [246] calls a vertex of a graph saturated if it is adjacent to every other vertex of the graph. He proves that every integral sum graph except K_{3} has at most two saturated vertices and gives the exact structure of all integral sum graphs that have exactly two saturated vertices. Chen [246] also proves that a connected integral sum graph with p > 1 vertices and q edges and no saturated vertices satisfies q ≤ p(3p − 2)/8 − 2. Wu, Mao, and Le [961] proved that mP_{n} are integral sum graphs. They also show that the conjecture of Harary [400] that the sum number of C_{n} equals the integral sum number of C_{n} if and only if n ≠ 3 or 5 is false and that for n ≠ 4 or 6 the integral sum number of C_{n} is at most 1.

He, Wang, Mi, Shen, and Yu [408] say that a graph has a tail if the graph contains a path for which each interior vertex has degree 2 and an end vertex of degree at least 3. They prove that every tree with a tail of length at least 3 is an integral sum graph.

B. Xu [970] has shown that the following are integral sum graphs: the union of any three stars; T ∪K_{1,n} for all trees T; mK_{3} for all m; and the union of any number of integral sum trees. Xu also proved that if 2G and 3G are integral sum graphs, then so is mG for all m > 1. Xu poses the question as to whether all disconnected forests are integral sum graphs. Nicholas and Somasundaram [689] prove that all banana trees (see Section 2.1 for the definition) and the union of any number of stars are integral sum graphs.

Liaw, Kuo, and Chang [599] proved that all caterpillars are integral sum graphs (see also [961] and [970] for some special cases of caterpillars). This shows that the assertion by Harary in [400] that K(1, 3) and S(2, 2) are not integral sum graphs is incorrect. They also prove that all cycles except C_{4} are integral sum graphs and they conjecture that every tree is an integral sum graph. Singh and Santhosh show that the crowns C_{n} ⊙ K_{1} are integral sum graphs for n ≥ 4 [840] and that the subdivision graphs of C_{n} ⊙ K_{1} are integral sum graphs for n ≥ 3 [748].

The integral sum number, ζ(G), of G is the minimum number of isolated vertices that must be added to G so that the resulting graph is an integral sum graph. Thus, by definition, G is a integral sum graph if and only if ζ(G) = 0. Harary [400] conjectured that ζ(K_{n}) = 2n − 3 for n ≥ 4. This conjecture was verified by Chen [243], by Sharary [801], and by B. Xu [970]. Yan and Liu proved: ζ(K_{n}−E(K_{r})) = n−1 when n ≥ 6, n ≡ 0 (mod 3) and r = 2n/3−1 [978]; ζ(K_{m.m}) = 2m−1 for m ≥ 2 [978]; ζ(K_{n} −edge) = 2n−4 for n ≥ 4 [978], [970]; if n ≥ 5 and n − 3 ≥ r, then ζ(K_{n} − E(K_{r})) ≥ n − 1 [978]; if ⌈2n/3⌉ − 1 > r ≥ 2, then ζ(K_{n} − E(K_{r})) ≥ 2n − r − 2 [978]; and if 2 ≤ m < n, and n = (i + 1)(im − i + 2)/2, then σ(K_{m,n}) = ζ(K_{m,n}) = (m − 1)(i + 1) + 1 while if (i + 1)(im − i + 2)/2 < n < (i + 2)[(i + 1)m − i + 1]/2, then σ(K_{m,n}) = ζ(K_{m,n}) = ⌈((m − 1)(i + 1)(i + 2) + 2n)/(2i + 2)⌉ [978].

Nagamochi, Miller, and Slamin [676] have determined upper and lower bounds on the sum number a graph. For most graphs G(V, E) they show that σ(G) = Ω(|E|). He, Yu, Mi, Sheng, and Wang [409] investigated ζ(K_{n} − E(K_{r})) where n ≥ 5 and r ≥ 2. They proved that ζ(K_{n} − E(K_{r})) = 0 when r = n or n − 1; ζ(K_{n} − E(K_{r})) = n − 2 when r = n−2; ζ(K_{n}−E(K_{r})) = n−1 when n−3 ≥ r ≥ ⌈2n/3⌉−1; ζ(K_{n}−E(K_{r})) = 3n−2r−4 when ⌈2n/3⌉ − 1 > r ≥ n/2; ζ(K_{n} − E(K_{r})) = 2n − 4 when ⌈2n/3⌉ − 1 ≥ n/2 > r ≥ 2. Moreover, they prove that if n ≥ 5, r ≥ 2, and r ≠ n − 1, then σ(K_{n} − E(K_{r})) = ζ(K_{n} − E(K_{r})). Dou and Gao [289] prove that for n ≥ 3, the fan F_{n} = P_{n} + K_{1} is an integral sum graph, ρ(F_{4}) = 1, ρ(F_{n}) = 2 for n ≠ 4, and σ(F_{4}) = 2, σ(F_{n}) = 3 for n = 3 or n ≥ 6 and n even, and σ(F_{n}) = 4 for n ≥ 6 and n odd.

Chen [243] has given some properties of integral sum labelings of graphs G with Δ(G) < |V (G)| − 1 whereas Nicholas, Somasundaram, and Vilfred [691] provided some general properties of connected integral sum graphs G with Δ(G) = |V (G)| − 1. They have shown that connected integral sum graphs G other than K_{3} with the property that G has exactly two vertices of maximum degree are unique and that a connected integral sum graph G other than K_{3} can have at most two vertices with degree |V (G)| − 1 (see also [935]).

Vilfred and Florida [932] have examined one-point unions of pairs of small complete graphs. They show that the one-point union of K_{3} and K_{2} and the one-point union of K_{3} and K_{3} are integral sum graphs whereas the one-point union of K_{4} and K_{2} and the one-point union of K_{4} and K_{3} are not integral sum graphs. In [933] Vilfred and Florida defined and investigated properties of maximal integral sum graphs.

Vilfred and Nicholas [936] have shown that the following graphs are integral sum graphs: banana trees, the union of any number of stars, fans P_{n} + K_{1} (n ≥ 2), Dutch windmills K_{3}^{(m)}, and the graph obtained by starting with any finite number of integral sum graphs G_{1}, G_{2}, …, G_{n} and any collections of n vertices with v_{i} ∈ G_{i} and creating a graph by identifying v_{1}, v_{2}, …, v_{n}. The same authors [937] also proved that G + v where G is a union of stars is an integral sum graph.

Melnikov and Pyatkin [649] have shown that every 2-regular graph except C_{4} is an integral sum graph and that for every positive integer r there exists an r-regular integral sum graph. They also show that the cube is not an integral sum graph. For any integral sum graph G, Melnikov and Pyatkin define the integral radius of G as the smallest natural number r(G) that has all its vertex labels in the interval [−r(G), r(G)]. For the family of all integral sum graphs of order n they use r(n) to denote maximum integral radius among all members of the family. Two questions they raise are: Is there a constant C such that r(n) ≤ C_{n} and for n > 2, is r(n) equal to the (n − 2)th prime?

The concepts of sum number and integral sum number have been extended to hypergraphs. Sonntag and Teichert [867] prove that every hypertree (i.e., every connected, non-trivial, cycle-free hypergraph) has sum number 1 provided that a certain cardinality condition for the number of edges is fulfilled. In [868] the same authors prove that for d ≥ 3 every d-uniform hypertree is an integral sum graph and that for n ≥ d + 2 the sum number of the complete d-uniform hypergraph on n vertices is d(n − d) + 1. They also prove that the integral sum number for the complete d-uniform hypergraph on n vertices is 0 when d = n or n − 1 and is between (d − 1)(n − d − 1) and d(n − d) + 1 for d ≤ n − 2. They conjecture that for d ≤ n − 2 the sum number and the integral sum number of the complete d-uniform hypergraph are equal. Teichert [908] proves that hypercycles have sum number 1 when each edge has cardinality at least 3 and that hyperwheels have sum number 1 under certain restrictions for the edge cardinalities. (A hypercycle C_{n} = (V_{n}, E_{n}) has V_{n} = ∪_{i=1}^{n}{v_{1}^{i}, v_{2}^{i}, …, v_{d_{i}−1}^{i}}, E_{n} = {e_{1}, e_{2}, …, e_{n}} with e_{i} = {v_{1}^{i}, …, v_{d_{i}}^{i} = v_{1}^{i+1}} where i + 1 is taken modulo n. A hyperwheel W_{n} = (V′_{n}, E′_{n}) has V′_{n} = V_{n} ∪ {c} ∪_{i=1}^{n} {v_{2}^{n+i}, …, v_{d_{n+i}−1}^{n+i}}, E′_{n} = E_{n} ∪ {e_{n+1}, …, e_{2n}} with e_{n+i} = {v_{1}^{n+i} = c, v_{2}^{n+i}, …, v_{d_{n+i}−1}^{n+i}, v_{d_{n+i}}^{n+i} = v_{1}^{i}}.)

Teichert [907] determined an upper bound for the sum number of the d-partite complete hypergraph K_{n_{1},...,n_{d}}^{d}. In [909] Teichert defines the strong hypercycle C_{n}^{d} to be the d-uniform hypergraph with the same vertices as C_{n} where any d consecutive vertices of C_{n} form an edge of C_{n}^{d}. He proves that for n ≥ 2d + 1 ≥ 5, σ(C_{n}^{d}) = d and for d ≥ 2, σ(C_{d+1}^{d}) = d. He also shows that σ(C_{5}^{3}) = 3; σ(C_{6}^{3}) = 2, and he conjectures that σ(C_{n}^{d}) < d for d ≥ 4 and d + 2 ≤ n ≤ 2d.

In [692] Nicholas and Vilfred define the edge reduced sum number of a graph as the minimum number of edges whose removal from the graph results in a sum graph. They show that for K_{n}, n ≥ 3, this number is (n(n−1)/2+⌊n/2⌋)/2. They ask for a characterization of graphs for which the edge reduced sum number is the same as its sum number. They conjecture that an integral sum graph of order p and size q exists if and only if q ≤ 3(p^{2} − 1)/8 − ⌊(p − 1)/4⌋ when p is odd and q ≤ 3(3p − 2)/8 when p is even. They also define the edge reduced integral sum number in an analogous way and conjecture that for K_{n} this number is (n − 1)(n − 3)/8 + ⌊(n − 1)/4⌋ when n is odd and n(n − 2)/8 when n is even.

For certain graphs G Vilfred and Florida [931] investigated the relationships among σ(G), ζ(G), χ(G), and χ′(G) where χ(G) is the chromatic number of G and χ′(G) is the edge chromatic number of G. They prove: σ(C_{4}) = ζ(C_{4}) > χ(C_{4}) = χ′(C_{4}); for n ≥ 3, ζ(C_{2n}) < σ(C_{2n}) = χ(C_{2n}) = χ′(C_{2n}); ζ(C_{2n+1}) < σ(C_{2n+1}) < χ(C_{2n+1}) = χ′(C_{2n+1}); for n ≥ 4, χ′(K_{n}) ≤ χ(K_{n}) < ζ(K_{n}) = σ(K_{n}); and for n ≥ 2, χ(P_{n}×P_{2}) < χ′(P_{n}×P_{2}) = ζ(P_{n} × P_{2}) = σ(P_{n} × P_{2}).

Alon and Scheinermann [42] generalized sum graphs by replacing the condition f(x)+ f(y) ∈ S with g(f(x), f(y)) ∈ S where g is an arbitrary symmetric polynomial. They called a graph with this property a g-graph and proved that for a given symmetric polynomial g not all graphs are g-graphs. On the other hand, for every symmetric polynomial g and every graph G there is some vertex labeling such that G together with at most |E(G)| isolated vertices is a g-graph.

Boland, Laskar, Turner, and Domke [189] investigated a modular version of sum graphs. They call a graph G(V, E) a mod sum graph (MSG) if there exists a positive integer n and an injective labeling from V to {1, 2, …, n − 1} such that xy ∈ E if and only if (f(x) + f(y)) (mod n) = f(z) for some vertex z. Obviously, all sum graphs are mod sum graphs. However, not all mod sum graphs are sum graphs. Boland et al. [189] have shown the following graphs are MSG: all trees on 3 or more vertices; all cycles on 4 or more vertices; and K_{2,n}. They further proved that K_{p} (p ≥ 2) is not MSG (see also [366]) and that W_{4} is MSG. They conjecture that W_{p} is MSG for p ≥ 4. This conjecture was refuted by Sutton, Miller, Ryan, and Slamin [899] who proved that for n ≠ 4, W_{n} is not MSG (the case where n is prime had been proved in 1994 by Ghoshal, Laskar, Pillone, and Fricke [366]. In the same paper Sutton et al. also showed that for n ≥ 3, K_{n,n} is not MSG. Ghoshal, Laskar, Pillone, and Fricke [366] proved that every connected graph is an induced subgraph of a connected MSG graph and any graph with n vertices and at least two vertices of degree n − 1 is not MSG.

Sutton, Miller, Ryan, and Slamin [899] define the mod sum number, ρ(G), of a connected graph G to be the least integer r such that G ∪ \overline{K_{r}} is MSG. Sutton and Miller [897] define the cocktail party graph H_{m,n}, m, n ≥ 2, as the graph with a vertex set V = {v_{1}, v_{2}, …, v_{mn}} partitioned into n independent sets V = {I_{1}, I_{2}, …, I_{n}} each of size m such that v_{i}v_{j} ∈ E for all i, j ∈ {1, 2, …, mn} where i ∈ I_{p}, j ∈ I_{q}, p ≠ q. The graphs H_{m,n} can be used to model relational database management systems (see [895]). Sutton and Miller prove that H_{m,n} is not MSG for n > m ≥ 3 and ρ(K_{n}) = n for n ≥ 4. In [896] Sutton, Draganova, and Miller prove that for n odd and n ≥ 5, ρ(W_{n}) = n and when n is even, ρ(W_{n}) = 2.

Dou and Gao [290] obtained exact values for ρ(K_{m,n}) and ρ(K_{m} − E(K_{n})) for some cases of m and n and bounds in the remaining cases. They call a graph G(V, E) a mod integral sum graph if there exists a positive integer n and an injective labeling from V to {0, 1, 2, …, n − 1} (note that 0 is included) such that xy ∈ E if and only if (f(x) + f(y)) (mod n) = f(z) for some vertex z. They define the mod integral sum number, ψ(G), of a connected graph G to be the least integer r such that G ∪ \overline{K_{r}} is a mod integral sum graph. They prove that for m + n ≥ 3, ψ(K_{m,n}) = ρ(K_{m,n}) and obtained exact values for ψ(K_{m} − E(K_{n})) for some cases of m and n and bounds in the remaining cases.

Wallace [938] has proved that K_{m,n} is MSG when n is even and n ≥ 2m or when n is odd and n ≥ 3m − 3 and that ρ(K_{m,n}) = m when 3 ≤ m ≤ n < 2m. He also proves that the complete m-partite K_{n_{1},n_{2},...,n_{m}} is not MSG when there exist n_{i} and n_{j} such that n_{i} < n_{j} < 2n_{i}. He poses the following conjectures: ρ(K_{m,n}) = n when 3m − 3 > n ≥ m ≥ 3; if K_{n_{1},n_{2},...,n_{m}} where n_{1} > n_{2} > ⋯ > n_{m}, is not MSG, then (m − 1)n_{m} ≤ ρ(K_{n_{1},n_{2},...,n_{m}}) ≤ (m − 1)n_{1}; if G has n vertices, then ρ(G) ≤ n; determining the mod sum number of a graph is NP -complete (Sutton has observed that Wallace probably meant to say ‘NP - hard’). Miller [652] has asked if it is possible for the mod sum number of a graph G be of the order |V (G)|^{2}.

In a sum graph G, a vertex w is called a working vertex if there is an edge uv in G such that w = u+v. If G = H ∪\overline{H_{r}} has a sum labeling such that H has no working vertex the labeling is called an exclusive sum labeling of H with respect G. The exclusive sum number, ϵ(H), of a graph H is the smallest integer r such that G ∪ \overline{K_{r}} has an exclusive sum labeling. The exclusive sum number is known in the following cases (see [656] and [660]): for n ≥ 3, ϵ(P_{n}) = 2; for n ≥ 3, ϵ(C_{n}) = 3; for n ≥ 3, ϵ(K_{n}) = 2n − 3; for n ≥ 4, ϵ(F_{n}) = n (fan of order n + 1); for n ≥ 4, ϵ(W_{n}) = n; ϵ(C_{3}^{(n)}) = 2n (friendship graph–see §2.2); m ≥ 2, n ≥ 2, ϵ(K_{m,n}) = m + n − 1; for n ≥ 2, S_{n} = n (star of order n + 1); ϵ(S_{m,n}) = max{m, n} (double star); H_{2,n} = 4n − 5 (cocktail party graph); and ϵ(caterpillar G) = Δ(G). Vilfred and Florida [934] proved that ϵ(P_{3} × P_{3}) = 4 and ϵ(P_{n} × P_{2}) = 3.

If ϵ(G) = Δ(G), then G is said to be an Δ-optimum summable graph. An exclusive sum labeling of a graph G using Δ(G) isolates is called a Δ-optimum exclusive sum labeling of G. Tuga, Miller, Ryan, and Ryjáček [919] show that some families of trees that are Δ-optimum summable and some that are not. They prove that if G is a tree that has at least one vertex that has two or more neighbors that are not leaves then ϵ(G) = Δ(G).

Grimaldi [393] has investigated labeling the vertices of a graph G(V, E) with n vertices with distinct elements of the ring Z_{n} so that xy ∈ E whenever (x + y)^{−1} exists in Z_{n}

In his 2001 Ph. D. thesis Sutton [895] introduced two methods of graph labelings with applications to storage and manipulation of relational database links specifically in mind. He calls a graph G = (V_{p} ∪ V_{i}, E) a sum* graph of G_{p} = (V_{p}, E_{p}) if there is an injective labeling λ of the vertices of G with non-negative integers with the property that uv ∈ E_{p} if and only if λ(u) + λ(v) = λ(z) for some vertex z ∈ G. The sum^{∗} number, σ^{∗}(G_{p}), is the minimum cardinality of a set of new vertices V_{i} such that there exists a sum* graph of G_{p} on the set of vertices V_{p} ∪ V_{i}. A mod sum* graph of G_{p} is defined in the identical fashion except the sum λ(u) + λ(v) is taken modulo n where the vertex labels of G are restricted to {0, 1, 2, …, n − 1}. The mod sum* number, ρ^{∗}(G_{p}), of a graph G_{p} is defined in the analogous way. Sum* graphs are a generalization of sum graphs and mod sum* graphs are a generalization of mod sum graphs. Sutton shows that every graph is an induced subgraph of a connected sum* graph. Sutton [895] poses the following conjectures: ρ(H_{m,n}) ≤ mn for m, n ≥ 2; σ^{∗}(G_{p}) ≤ |V_{p}|; and ρ^{∗}(G_{p}) ≤ |V_{p}|.

The following table summarizing what is known about sum graphs, mod sum graphs, sum* graphs, and mod sum* graphs is reproduced from Sutton’s Ph. D. thesis [895]. It was updated by J. Gallian in 2006. A question mark indicates the value is unknown. The results on sum* and mod sum* graphs are found in [895].

Table 19: Summary of Sum Graph Labelings

Graph  σ(G)  ρ(G) σ^{∗}(G) ρ^{∗}(G)
K_{2} = S_{1}  1  1  0  0
stars, S_{n}, n ≥ 2  1  0  0  0
trees T_{n}, n ≥ 3 when T_{n} ≠ S_{n}  1  0  1  0
C_{3}  2  0  1  0
C_{4}  3  0  2  0
C_{n}, n > 4  2  0  2  0
W_{4}  4  0  2  0
W_{n}, n ≥ 5, n odd  n  n  2  0
W_{n}, n ≥ 6, n even  \frac{n}{2} + 2  2  2  0
fan, F_{4},  2  1  1  0
fans, F_{n}, n ≥ 5, n odd  ?  2  1  0
fans, F_{n}, n ≥ 6, n even  3  2  1  0
K_{n}, n ≥ 3  2n − 3  n n − 2  0
cocktail party graphs, H_{2,n}  4n − 5  0  ?  0
C_{n}^{(t)} (n, t) ≠ (4, 1) (see §2.2)  2  ?  ?  ?
K_{n,n}  ⌈\frac{4n−3}{2}⌉  ?  ?  ?
K_{m,n}, 2nm ≥ n ≥ 3  ?  n  ?  ?
K_{m,n} m ≥ 3n − 3, n ≥ 3, m odd  ?  0  ?  0
K_{m,n}, m ≥ 2n, n ≥ 3, m even  ?  0  ?  0
K_{m,n}, m < n  ⌈(kn − k)/2 + m/(k − 1)⌉ ?  ?  ?

k = ⌈\sqrt{1 + (8m + n − 1)(n − 1)/2} ⌉

K_{n,n} − E(nK_{2}), n ≥ 6  2n − 3  n − 2  ?  ?

the electronic journal of combinatorics 16 (2009), #DS6 135

7.2 Prime and Vertex Prime Labelings

The notion of a prime labeling originated with Entringer and was introduced in a paper by Tout, Dabboucy, and Howalla [912]. A graph with vertex set V is said to have a prime labeling if its vertices are labeled with distinct integers 1, 2, …, |V | such that for each edge xy the labels assigned to x and y are relatively prime. Around 1980, Entringer conjectured that all trees have a prime labeling. So far, there has been little progress towards proving this conjecture. Among the classes of trees known to have prime labelings are: paths, stars, caterpillars, complete binary trees, spiders (i.e., trees with one vertex of degree at least 3 and with all other vertices with degree at most 2), olive trees (i.e., a rooted tree consisting of k branches such that the ith branch is a path of length i), and all trees of order up to 50 (see [704], [706], [912] and [339]).

Other graphs with prime labelings include all cycles and the disjoint union of C_{2k} and C_{n} [277]. The complete graph K_{n} does not have a prime labeling for n ≥ 4 and W_{n} is prime if and only if n is even (see [594]).

Seoud, Diab, and Elsakhawi [761] have shown the following graphs are prime: fans; helms; flowers (see §2.2); stars; K_{2,n}; and K_{3,n} unless n = 3 or 7. They also shown that P_{n} + \overline{K_{m}} (m ≥ 3) is not prime. Tout, Dabboucy, and Howalla [912] proved that C_{m} ⊙ \overline{K_{n}} is prime for all m and n.

For m and n at least 3, Seoud and Youssef [767] define S_{n}^{(m)}, the (m, n)-gon star, as the graph obtained from the cycle C_{n} by joining the two end vertices of the path P_{m−2} to every pair of consecutive vertices of the cycle such that each of the end vertices of the path is connected to exactly one vertex of the cycle. Seoud and Youssef [767] have proved the following graphs have prime labelings: books; S_{n}^{(m)}; C_{n} ⊙ P_{m}; P_{n} + \overline{K}_{2} if and only if n = 2 or n is odd; and C_{n} ⊙ K_{1} with a complete binary tree of order 2^{k} − 1 (k ≥ 2) attached at each pendant vertex. They also prove that every spanning subgraph of a prime graph is prime and every graph is a subgraph of a prime graph. They conjecture that all unicycle graphs have prime labelings. Seoud and Youssef [767] proved the following graphs are not prime: C_{m} + C_{n}; C_{n}^{2} for n ≥ 4; P_{n}^{2} for n = 6 and for n ≥ 8; and Möbius ladders M_{n} for n even (see §2.3 for the definition). They also give an exact formula for the maximum number of edges in a prime graph of order n and an upper bound for the chromatic number of a prime graph.

Youssef and Elsakhawi [1001] have shown: the union of stars S_{m} ∪ S_{n}, are prime; the union of cycles and stars C_{m} ∪ S_{n} are prime; K_{m} ∪ P_{n} is prime if and only if m is at most 3 or if m = 4 and n is odd; K_{n} ⊙ K_{1} is prime if and only if n ≤ 7; K_{n} ⊙ \overline{K_{2}} is prime if and only if n ≤ 16; 6K_{m} ∪ S_{n} is prime if and only if the number of primes less than or equal to m + n + 1 is at least m; and that the complement of every prime graph with order at least 20 is not prime.

Salmasian [742] has shown that every tree with n vertices (n ≥ 50) can be labeled with n integers between 1 and 4n such that every two adjacent vertices have relatively prime labels. Pikhurko [706] has improved this by showing that for any c > 0 there is an N such that any tree of order n > N can be labeled with n integers between 1 and (1 + c)n such that labels of adjacent vertices are relatively prime.

Varkey and Singh (see [922]) have shown the following graphs have prime labelings: ladders, crowns, cycles with a chord, books, one point unions of C_{n}, and L_{n} + K_{1}. Varkey [922] has shown that graph obtained by connecting two points with internally disjoint paths of equal length are prime. Varkey defines a twig as a graph obtained from a path by attaching exactly two pendent edges to each internal vertex of the path. He proves that twigs obtained from a path of odd length (at least 3) and lotus inside a circle (see §5.1 for the definition) graphs are prime.

Baskar Babujee and Vishnupriya [116] proved the following graphs have prime labelings: nP_{2}, P_{n} ∪ P_{n} ∪ ⋯ P_{n}, bistars (that is, the graphs obtained by joining the centers of two identical stars with an edge), and the graph obtained by subdividing the edge joining edge of a bistar. Baskar Babujee [113] obtained prime labelings for the graphs: (P_{m} ∪ nK_{1}) + \overline{K_{2}}, (C_{m} ∪ nK_{1}) + \overline{K_{2}}, (P_{m} ∪ C_{n} ∪ \overline{K}_{r}) + \overline{K_{2}}, C_{n} ∪ C_{n+1}, (2n − 2)C_{2n} (n > 1), C_{n} ∪ mP_{k} and the graph obtained by subdividing each edge of a star once.

Given a collection of graphs G_{1}, …, G_{n} and some fixed vertex v_{i} from each G_{i}, Lee, Wui, and Yeh [594] define Amal{(G_{i}, v_{i})}, the almagamation of {(G_{i}, v_{i})| i = 1, …, n}, as the graph obtained by taking the union of the G_{i} and identifying v_{1}, v_{2}, …, v_{n}. Lee, Wui, and Yeh [594] have shown Amal{(G_{i}, v_{i})} has a prime labeling when G_{i} are paths and when G_{i} are cycles. They also showed that the almagamation of any number of copies of W_{n}, n odd, with a common vertex is not prime. They conjecture that for any tree T and any vertex v from T, the almagamation of two or more copies of T with v in common is prime. They further conjecture that the almagamation of two or more copies of W_{n} that share a common point is prime when n is even (n ≠ 4). Vilfred, Somasundaram, and Nicholas [928] have proved this conjecture for the case that n ≡ 2 (mod 4) where the central vertices are identified.

Vilfred, Somasundaram, and Nicholas [928] have also proved the following: helms are prime; the grid P_{m} ×P_{n} is prime when m ≤ 3 and n is a prime greater than m; the double cone C_{n} + \overline{K_{2}} is prime only for n = 3; the double fan P_{n} × \overline{K_{2}} (n ≠ 2) is prime if and only if n is odd or n = 2; and every cycle with a P_{k}-chord is prime. They conjecture that the grid P_{m} × P_{n} is prime when n is prime and n > m. This conjecture was proved by Sundaram, Ponraj, and Somasundaram [892]. In the same article they also showed that P_{n} × P_{n} is prime when n is prime.

For any finite collection {G_{i}, u_{i}v_{i}} of graphs G_{i}, each with a fixed edge u_{i}v_{i}, Carlson [224] defines the edge amalgamation Edgeamal{(G_{i}, u_{i}v_{i})} as the graph obtained by taking the union of all the G_{i} and identifying their fixed edges. The case where all the graphs are cycles she calls generalized books. She proves that all generalized books are prime graphs. Moreover, she shows that graphs obtained by taking the union of cycles and identifying in each cycle the path P_{n} are also prime. Carlson also proves that C_{m}-snakes are prime (see §2.2) for the definition).

Yao, Cheng, Zhongfu, and Yao [989] have shown: a tree of order p with maximum degree at least p/2 is prime; a tree of order p with maximum degree at least p/2 has a vertex subdivision that is prime; if a tree T has an edge u_{1}u_{2} such that the two components T_{1} and T_{2} of T − u_{1}u_{2} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2 and d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, then T is prime when |T_{1}| + |T_{2}| is prime; if a tree T has two edges u_{1}u_{2} and u_{2}u_{3} such that the three components T_{1}, T_{2}, and T_{3} of T − {u_{1}u_{2}, u_{2}u_{3}} have the properties that d_{T_{1}}(u_{1}) ≥ |T_{1}|/2, d_{T_{2}}(u_{2}) ≥ |T_{2}|/2, and d_{T_{3}}(u_{3}) ≥ |T_{3}|/2, then T is prime when |T_{1}| + |T_{2}| + |T_{3}| is prime.

A dual of prime labelings has been introduced by Deretsky, Lee, and Mitchem [277]. They say a graph with edge set E has a vertex prime labeling if its edges can be labeled with distinct integers 1, …, |E| such that for each vertex of degree at least 2 the greatest common divisor of the labels on its incident edges is 1. Deretsky, Lee, and Mitchem show the following graphs have vertex prime labelings: forests; all connected graphs; C_{2k} ∪ C_{n}; C_{2m} ∪ C_{2n} ∪ C_{2k+1}; C_{2m} ∪ C_{2n} ∪ C_{2t} ∪ C_{k}; and 5C_{2m}. They further prove that a graph with exactly two components, one of which is not an odd cycle, has a vertex prime labeling and a 2-regular graph with at least two odd cycles does not have a vertex prime labeling. They conjecture that a 2-regular graph has a vertex prime labeling if and only if it does not have two odd cycles. Let G = ⋃_{i=1}^{t} C_{2n_{i}} and N = ∑_{i=1}^{t} n_{i}. In [191] Borosh, Hensley and Hobbs proved that there is a positive constant n_{0} such that the conjecture of Deretsky et al. is true for the following cases: G is the disjoint union of at most seven cycles; G is a union of cycles all of the same even length 2n where n ≤ 150 000 or where n ≥ n_{0}; n_{i} ≥ (log N)^{4 log log log n} for all i = 1, …, t; and when each C_{2n_{i}} is repeated at most n_{i} times. They end their paper with a discussion of graphs whose components are all even cycles, and of graphs with some components that are not cycles and some components that are odd cycles.

The tables following summarize the state of knowledge about prime labelings and vertex prime labelings. In the table P means prime labeling exists VP means vertex prime labeling exists.

A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property.

Table 20: Summary of Prime Labelings

Graph  Types Notes
P_{n}  P  [339]
stars  P  [339]
caterpillars  P  [339]
complete binary trees P  [339]
spiders  P  [339]
trees  P?  [594]
C_{n}  P  [277]
C_{n} ∪ C_{2m}  P  [277]
K_{n}  P  iff n ≤ 3 [594]
W_{n}  P  iff n is even [594]
helms  P  [761], [928]
fans  P  [761]
flowers  P  [761]
K_{2,n}  P  [761]
K_{3,n}  P  n ≠ 3, 7 [761]
P_{n} + \overline{K_{m}}  not P n ≥ 3 [761]
P_{n} + \overline{K_{2}}  P  iff n = 2 or n is odd [761]

Table 20: Summary of Prime Labelings continued

Graph  Types Notes
books  P  [767]
C_{n} ⊙ P_{m}  P  [767]
unicyclic graphs  P?  [767]
C_{m} + C_{n}  not P [767]
C_{n}^{2}  not P n ≥ 4 [767]
P_{n}^{2}  not P n ≥ 6, n ≠ 7 [767]
M_{n} (Möbius ladders) not P n even [767]
S_{m} ∪ S_{n}  P  [1001]
C_{m} ∪ S_{n}  P  [1001]
K_{m} ∪ S_{n}  P  iff number of primes ≤ m + n + 1

is at least m [1001]

K_{n} · K_{1}  P  iff n ≤ 7 [1001]
P_{n} × P_{2} (ladders)  P  [922]
P_{m} × P_{n} (grids)  P  m ≤ 3, m > n, n prime [928]
C_{n} ⊙ K_{1} (crowns)  P  [922]
cycles with a chord P  [922]

Graph  Types Notes
wheels  P  [922]
C_{n} ⊙ \overline{K_{2}}  P  iff n = 3 [928]
P_{n} ⊙ \overline{K_{2}}  P  iff n ≠ 2 [928]
C_{m}-snakes (see §2.2) P  [224]
unicyclic  P?  [761]
C_{m} ⊙ P_{n}  P  [767]
K_{1,n} + \overline{K_{2}}  P  [842]
K_{1,n} + K_{2}  P  n prime, n ≥ 4 [842]
P_{n} ⊙ K_{1} (combs)  P  n ≥ 2 [842]
P_{1} ∪ P_{2} ∪ ⋯ ∪ P_{n}  P  [842]
P_{n} × P_{2} (ladders)  P  n ≥ 3, 2n + 1 prime [842]
P?  n ≥ 3 [842]
C_{m}^{(n)} (see §2.2  P  n(m − 1) + 1 prime [842]
triangular snakes  P  [842]
quadrilateral snakes P  [842]

Table 21: Summary of Vertex Prime Labelings

Graph  Types Notes
C_{m} + C_{n}  not P [767]
C_{n}^{2}  not P n ≥ 4 [767]
P_{n}  not P n = 6, n ≥ 8 [767]
M_{2n} (Möbius ladders) not P [767]
connected graphs  VP  [277]
forests  VP  [277]
C_{2m} ∪ C_{n}  VP  [277]
C_{2m} ∪ C_{2n} ∪ C_{2k+1}  VP  [277]
C_{2m} ∪ C_{2n} ∪ C_{2t} ∪ C_{k} VP  [277]
5C_{2m}  VP  [277]
G ∪ H  VP  if G, H are connected and

one is not an odd cycle [277]

2-regular graph G  not VP G has at least 2 odd cycles [277]
VP?  iff G has at most 1 odd cycle [277]

7.3 Edge-graceful Labelings

In 1985, Lo [623] introduced the notion of edge-graceful graphs. A graph G(V, E) is said to be edge-graceful if there exists a bijection f from E to {1, 2, …, |E|} such that the induced mapping f^{+} from V to {0, 1, …, |V | − 1} given by f^{+}(x) = (∑ f(xy)) (mod |V |) taken over all edges xy is a bijection. Note that an edge-graceful graph is antimagic (see §6.1). A necessary condition for a graph with p vertices and q edges to be edge-graceful is that q(q + 1) ≡ p(p + 1)/2 (mod p). Lee [533] notes that this necessary condition extends to any multigraph with p vertices and q edges. It was conjectured by Lee [533] that any connected simple (p, q)-graph with q(q + 1) ≡ p(p − 1)/2 (mod p) vertices is edge-graceful. Lee, Kitagaki, Young, and Kocay [536] prove that the conjecture is true for maximal outerplanar graphs. Lee and Murthy [530] proved that K_{n} is edge-graceful if and only if n ≢ 2 (mod 4). (An edge-graceful labeling given in [623] for K_{n} for n ≢ 2 (mod 4) is incorrect.) Lee [533] notes that a multigraph with p ≡ 2 (mod 4) vertices is not edge-graceful and conjectures that this condition is sufficient for the edge-gracefulness of connected graphs. Lee [532] has conjectured that all trees of odd order are edgegraceful. Small [854] has proved that spiders for which every vertex has odd degree with the property that the distance from the vertex of degree greater than 2 to each end vertex is the same are edge-graceful. Keene and Simoson [484] proved that all spiders of odd order with exactly three end vertices are edge-graceful. Cabaniss, Low, and Mitchem [208] have shown that regular spiders of odd order are edge-graceful.

Lee and Seah [565] have shown that K_{n,n,...,n} is edge-graceful if and only if n is odd and the number of partite sets is either odd or a multiple of 4. Lee and Seah [564] have also proved that C_{n}^{k} (the kth power of C_{n}) is edge-graceful for k < ⌊n/2⌋ if and only if n is odd and C_{n}^{k} is edge-graceful for k ≥ ⌊n/2⌋ if and only if n ≢ 2 (mod 4) (see also [208]). Lee, Seah, and Wang [570] gave a complete characterization of edge-graceful P_{n}^{k} graphs. Shiu, Lam, and Cheng [814] proved that the composition of the path P_{3} and any null graph of odd order is edge-graceful.

Lo proved that all odd cycles are edge-graceful and Wilson and Riskin [956] proved the Cartesian product of any number of odd cycles is edge-graceful. Lee, Ma, Valdes, and Tong [546] investigated the edge-gracefulness of grids P_{m} × P_{n}. The necessity condition of Lo [623] that a (p, q) graph must satisfy q(q + 1) ≡ 0 or p/2 (mod p) severely limits the possibilities. Lee et al. prove the following: P_{2} × P_{n} is not edge-graceful for all n > 1; P_{3} × P_{n} is edge-graceful if and only if n = 1 or n = 4; P_{4} × P_{n} is edge-graceful if and only if n = 3 or n = 4; P_{5} × P_{n} is edge-graceful if and only if n = 1; P_{2m} × P_{2n} is edge-graceful if and only if m = n = 2. They conjecture that for all m, n ≥ 10 of the form m = (2k + 1)(4k + 1), n = (2k + 1)(4k + 3), the grids P_{m} × P_{n} are edge-graceful. Riskin and Weidman [729] proved: if G is an edge-graceful 2r-regular graph with p vertices and q edges and (r, kp) = 1, then kG is edge-graceful when k is odd; when n and k are odd, kC_{n}^{r} is edge-graceful; and if G is the cartesian product of an odd number of odd cycles and k is odd, then kG is edge-graceful. They conjecture that the disjoint union of an odd number of copies of a 2r-regular edge-graceful graph is edge-graceful.

Shiu, Lee, and Schaffer [820] investigated the edge-gracefulness of multigraphs derived from paths, combs, and spiders obtained by replacing each edge by k parallel edges. Lee, Ng, Ho, and Saba [553] construct edge-graceful multigraphs starting with paths and spiders by adding certain edges to the original graphs. Lee and Seah [566] have also investigated edge-gracefulness of various multigraphs.

Lee and Seah (see [533]) define a sunflower graph SF (n) as the graph obtained by starting with an n-cycle with consecutive vertices v_{1}, v_{2}, …, v_{n} and creating new vertices w_{1}, w_{2}, …, w_{n} with w_{i} connected to v_{i} and v_{i+1} (v_{n+1} is v_{1}). In [567] they prove that SF (n) is edge-graceful if and only if n is even. In the same paper they prove that C_{3} is the only triangular snake that is edge-graceful. Lee and Seah [564] prove that for k ≤ n/2, C_{n}^{k} is edge-graceful if and only if n is odd, and for k ≥ n/2, C_{n}^{k} is edge-graceful if and only if n ≢ 2 (mod 4). Lee, Seah, and Lo (see [533]) have proved that for n odd, C_{2n} ∪ C_{2n+1}, C_{n} ∪ C_{2n+2}, and C_{n} ∪ C_{4n} are edge-graceful. They also show that for odd k and odd n, kC_{n} is edge-graceful. Lee and Seah (see [533]) prove that the generalized Petersen graph P (n, k) (see Section 2.7 for the definition) is edge-graceful if and only if n is even and k < n/2. In particular, P (n, 1) = C_{n} × P_{2} is edge-graceful if and only if n is even.

Schaffer and Lee [750] proved that C_{m} × C_{n} (m > 2, n > 2) is edge-graceful if and only if m and n are odd. They also showed that if G and H are edge-graceful regular graphs of odd order then G × H is edge-graceful and that if G and H are edge-graceful graphs where G is c-regular of odd order m and H is d-regular of odd order n, then G×H is edge-magic if gcd(c, n) = gcd(d, m) = 1. They further show that if H has odd order, is 2d-regular and edge-graceful with gcd(d, m) = 1, then C_{2m} × H is edge-magic, and if G is odd-regular, edge-graceful of even order m that is not divisible by 3, and G can be partitioned into 1-factors, then G × C_{m} is edge-graceful.

In 1987 Lee (see [568]) conjectured that C_{2m} ∪ C_{2n+1} is edge-graceful for all m and n except for C_{4} ∪ C_{3}. Lee, Seah, and Lo [568] have proved this for the case that m = n and m is odd. They also prove: the disjoint union of an odd number copies of C_{m} is edge-graceful when m is odd; C_{n} ∪ C_{2n+2} is edge-graceful; and C_{n} ∪ C_{4n} is edge-graceful for n odd.

Kendrick and Lee (see [533]) proved that there are only finitely many n for which K_{m,n} is edge-graceful and they completely solve the problem for m = 2 and m = 3. Ho, Lee, and Seah [437] use S(n; a_{1}, a_{2}, …, a_{k}) where n is odd and 1 ≤ a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{k} < n/2 to denote the (n, nk)-multigraph with vertices v_{0}, v_{1}, …, v_{n−1} and edge set {v_{i}v_{j}| i ≠ j, i − j ≡ a_{t} (mod n) for t = 1, 2, …, k}. They prove that all such multigraphs are edge-graceful. Lee and Pritikin (see [533]) prove that the Möbius ladders (see §2.2 for definition) of order 4n are edge-graceful. Lee, Tong, and Seah [577] have conjectured that the total graph of a (p, p)-graph is edge-graceful if and only if p is even. They have proved this conjecture for cycles.

Kuang, Lee, Mitchem, and Wang [520] have conjectured that unicyclic graphs of odd order are edge-graceful. They have verified this conjecture in the following cases: graphs obtained by identifying an end point of a path P_{m} with a vertex of C_{n} when m + n is even; crowns with one pendant edge deleted; graphs obtained from crowns by identifying an endpoint of P_{m}, m odd, with a vertex of degree 1; amalgamations of a cycle and a star obtained by identifying the center of the star with a cycle vertex where the resulting graph has odd order; graphs obtained from C_{n} by joining a pendant edge to n − 1 of the cycle vertices and two pendant edges to the remaining cycle vertex.

Gayathri and Subbiah [360] say a graph G(V, E) has a strong edge gracefulabeling if there is an injection f from the E to {1, 2, 3, …, ⟨3|E|/2⟩} such that the induced mapping f^{+} from V defined by f^{+}(u) = (Σf(uv)) (mod 2|V |) taken all edges uv is an injection. They prove the following graphs have strong edge graceful labelings: P_{n}(n ≥ 3), C_{n}, K_{1,n}(n ≥ 2), crowns C_{n} ⊕ K_{1}, and fans P_{n} + K_{1}(n ≥ 2).

Hefetz [412] has shown that a graph G = (V, E) of the form G = H ∪ f_{1} ∪ f_{2} ∪ ⋯ ∪ f_{r} where H = (V, E′) is edge-graceful and the f_{i}’s are 2-factors is also edge-graceful and that a regular graph of even degree that has a 2-factor consisting of k cycles each of length t where k and t are odd is edge-graceful.

Bača and Holländer [90] investigated a generalization of edge-graceful labeling called (a, b)-consecutive labelings. A connected graph G(V, E) is said to have an (a, b)-consecutive labeling where a is a nonnegative integer and b is a positive proper divisor of |V |, if there is a bijection from E to {1, 2, …, |E|} such that if each vertex v is assigned the sum of all edges incident to v the vertex labels are distinct and they can be partitioned into |V |/b intervals

W_{j} = [wmin = (j − 1)b + (j − 1)a, wmin + jb + (j − 1)a − 1],

where 1 ≤ j ≤ p/b and wmin is the minimum value of the vertices. They present necessary conditions for (a, b)-consecutive labelings and describe (a, b)-consecutive labelings of the generalized Petersen graphs for some values of a and b.

A graph with p vertices and q edges is said to be k-edge-graceful if its edges can be labeled with k, k + 1, …, k + q − 1 such that the sums of the edges incident to each vertex are distinct modulo p. In [580] Lee and Wang show that for each k ≠ 1 there are only finitely many trees that are k-edge graceful (there are infinitely many 1-edge graceful trees). They describe completely the k-edge-graceful trees for k = 0, 2, 3, 4, and 5.

In 1991 Lee [533] defined the edge-graceful spectrum of a graph G as the set of all nonnegative integers k such that G has a k-edge graceful labeling. In [584] Lee, Wang, Ng, and Wang determine the edge-graceful spectrum of the following graphs: G ⊙ K_{1} where G is an even cycle with one chord; two even cycles of the same order joined by an edge; and two even cycles of the same order sharing a common vertex with an arbitrary number of pendant edges attached at the common vertex (butterfly graph). Lee, Chen, and Wang [535] have determined the edge-graceful spectra for various cases of cycles with a chord and for certain cases of graphs obtained by joining two disjoint cycles with an edge (i.e., dumbbell graphs). Shiu, Ling, and Low [822] found the the entire edge-graceful spectra of cycles with one chord. Kang, Lee and Wang [471] determined the edge-graceful spectra of wheels and Wang, Hsiao and Lee [947] determined the edge-graceful spectra of the square of P_{n} for odd n (see also Lee, Wang, and Hsiao [582]).

Lee, Levesque, Lo, and Schaffer [542] investigate the edge-graceful spectra of cylinders. They prove: for odd n ≥ 3 and m ≡ 2 (mod) 4, the spectra of C_{n} × P_{m} is ∅; for m = 3 and m ≡ 0, 1 or 3 (mod 4), the spectra of C_{4} × P_{m} is ∅; for even n ≥ 4, the spectra of C_{n} × P_{2} is all natural numbers; the spectra of C_{n} × P_{4} is all odd positive integers if and only if n ≡ 3 (mod) 4; and C_{n} × P_{4} is all even positive integers if and only if n ≡ 1 (mod)

4. They conjecture that C_{4} × P_{m} is k-edge-graceful for some k if and only if m ≡ 2 (mod)

4. Shiu, Ling, and Low [823] determine the edge-graceful spectra of all connected bicyclic graphs without pendant edges.

A graph G(V, E) is called super edge-graceful if there is a bijection f from E to {0, ±1, ±2, …, ±(|E| − 1)/2} when |E| is odd and from E to {±1, ±2, …, ±|E|/2} when |E| is even such that the induced vertex labeling f^{∗} defined by f^{∗}(u) = Σf(uv) over all edges uv is a bijection from V to {0, ±1, ±2, …, ±(p − 1)/2} when p is odd and from V to {±1, ±2, …, ±p/2} when p is even. Lee, Wang, and Nowak [585] proved the following: K_{1,n} is super-edge-magic if and only if n is even; the double star DS(m, n) (that is, the graph obtained by joining the centers of K_{1,m} and K_{1,n} by an edge) is super edge-graceful if and only if m and n are both odd. They conjecture that all trees of odd order are super edge-graceful.

Shiu [809] has shown that C_{n} ×P_{2} is super-edge-graceful for all n ≥ 2. More generally, he defines a family of graphs that includes C_{n} × P_{2} and generalized Petersen graphs are follows. For any permutation θ on n symbols without a fixed point the θ-Petersen graph P (n; θ) is the graph with vertex set {u_{1}, u_{2}, …, u_{n}} ∪ {v_{1}, v_{2}, …, v_{n}} and edge set {u_{i}u_{i+1}, u_{i}w_{i}, w_{i}w_{θ(i)} | 1 ≤ i ≤ n} where addition of subscripts is done modulo n. (The graph P (n; θ) need not be simple.) Shiu proves that P (n; θ) is super-edge-graceful for all n ≥ 2. He also shows that certain other families of connected cubic multigraphs are super-edge-graceful and conjectures that every connected cubic of multigraph except K_{4} and the graph with 2 vertices and 3 edges is super-edge-graceful.

In [812] Shiu and Lam investigated the super-edge-gracefulness of fans and wheel-like graphs. They showed that fans F_{2n} and wheels W_{2n} are super-edge-graceful. Although F_{3} and W_{3} are not super-edge-graceful the general cases F_{2n+1} and W_{2n+1} are open. For a positive integer n_{1} and even positive integers n_{2}, n_{3}, …, n_{m} they define an m-level wheel as follows. A wheel is a 1-level wheel and the cycle of the wheel is the 1-level cycle. An i-level wheel is obtained from an (i−1)-level wheel by appending n_{i}/2 pairs of edges from any number of vertices of the i − 1-level cycle to n_{i} new vertices that form the vertices in the i-level cycle. They prove that all m-level wheels are super-edge-graceful. They also prove that for n odd C_{m} ⊙ \overline{K_{n}} is super-edge-graceful, for odd m ≥ 3 and even n ≥ 2 C_{m} ⊙ \overline{K_{n}} is edge-graceful, and for m ≥ 2 and n ≥ 1 C_{m} ⊙ \overline{K_{n}} is super-edge-graceful. For a cycle C_{m} with consecutive vertices v_{1}, v_{2}, …, v_{m} and nonnegative integers n_{1}, n_{2}, …, n_{m} they define the graph A(m; n_{1}, n_{2}, …, n_{m}) as the graph obtained from C_{m} by attaching n_{i} edges to the vertex v_{i} for 1 ≤ i ≤ m. They prove A(m; n_{1}, n_{2}, …, n_{m}) is super-edge-graceful if m is odd and A(m; n_{1}, n_{2}, …, n_{m}) is super-edge-graceful if m is even and all the n_{i} are positive and have the same parity. Chung, Lee, Gao, and Schaffer [261] provide super edge-graceful labelings for various even order paths, spiders and disjoint unions of two stars. In [260] Chung and Lee characterize spiders of even orders that are not superedge-graceful and exhibit some spiders of even order of diameter at most four that are super-edge-graceful. They raised the question of which paths super edge-graceful. This was answered by Cichacz, Fronček and Xu [262] who showed that the only paths that are not super edge-graceful are P_{2} and P_{4}.

Although it is not the case that a super edge-graceful graph is edge-graceful, Lee, Chen, Yera, and Wang [534] proved that if G is a super edge-graceful with p vertices and q edges and q ≡ −1 (mod p) when q is even, or q ≡ 0 (mod p) when q is odd, then G is also edge-graceful. They also prove: the graph obtained from a connected super edge-graceful unicyclic graph of even order by joining any two nonadjacent vertices by an edge is super edge-graceful; the graph obtained from a super edge-graceful graph with p vertices and p + 1 edges by appending two edges to any vertex is super edge-graceful; and the one-point union of two identical cycles is super edge-graceful.

Gayathri, Duraisamy, and Tamilselvi [357] calls a (p, q)-graph with q ≥ p even edgegraceful if there is an injection f from the set of edges to {1, 2, 3, …, 2q} such that the values of the induced mapping f^{+} from the vertex set to {0, 1, 2, …, 2q − 1} given by f^{+}(x) = (Σf(xy))(mod 2q) over all edges xy are distinct and even. In [357] and [356] Gayathri et al. prove the following: cycles are even edge-graceful if and only if the cycles are odd; even cycles with one pendant edge are even edge-graceful; wheels are even edgegraceful; gears (see §2.2 for the definition) are not even edge-graceful; fans P_{n} + K_{1} are even edge-graceful; C_{4} ∪ P_{m} for all m are even edge-graceful; C_{2n+1} ∪ P_{2n+1} are even edgegraceful; crowns C_{n} ⊙K_{1} are even edge-graceful; C_{n}^{(m)} (see §2.2 for the definition) are even edge-graceful; sunflowers (see §3.7 for the definition) are even edge-graceful; triangular snakes (see §2.2 for the definition) are even edge-graceful; closed helms (see §2.2 for the definition) with the center vertex removed are even edge-graceful; graphs decomposable into two odd Hamiltonian cycles are even edge-graceful; and odd order graphs that are decomposable into three Hamiltonian cycles are even edge-graceful.

In [356] Gayathri and Duraisamy generalized the definition of even edge-graceful to include (p, q)-graphs with q < p by changing the modulus from 2q the maximum of 2q and 2p. With this version of the definition, they have shown that trees of even order are not even edge-graceful whereas, for odd order graphs, the following are even edge-graceful: banana trees (see §2.1 for the definition); graphs obtained joining the centers of two stars by a path; P_{n} ⊙ K_{1,m}; graphs obtained by identifying an end point from each of any number of copies of P_{3} and P_{2}; bistars (that is, graphs obtained by joining the centers of two stars with an edge); and graphs obtained by appending the end point of a path to the center of a star. They define odd edge-graceful graphs in the analogous way and provide a few results about such graphs.

Lee, Pan, and Tsai [556] call a graph G with p vertices and q edges vertex-graceful if there exists a labeling f V (G) → {1, 2, …, p} such that the induced labeling f^{+} from E(G) to Z_{q} defined by f^{+}(uv) = f(u) + f(v) (mod q) is a bijection. Vertex-graceful graphs can be viewed the dual of edge-graceful graphs. They call a vertex-graceful graph strong vertex-graceful if the values of f^{+}(E(G) are consecutive. They observe that the class of vertex-graceful graphs properly contains the super edge-magic graphs and strong vertexgraceful graphs are super edge-magic. They provide vertex-graceful and strong vertexgraceful labelings for various (p, p + 1)-graphs of small order and their amalgamations.

As a dual to super edge-graceful graphs Lee and Wei [587] define a graph G(V, E) to be super vertex-graceful if there is a bijection f from V to {±1, ±2, …, ±(|V | − 1)/2} when |V | is odd and from V to {±1, ±2, …, ±|V |/2} when |V | is even such that the induced edge labeling f^{∗} defined by f^{+}(uv) = f(u)+f(v) over all edges uv is a bijection from E to {0, ±1, ±2, …, ±(|E| − 1)/2} when |E| is odd and from E to {±1, ±2, …, ±|E|/2} when |E| is even. They show: for m and n_{1}, n_{2}, …, n_{m} each at least 3, P_{n_{1}} × P_{n_{2}} × ⋯ × P_{n_{m}} is not super vertex-graceful; for n odd, books K_{1,n} × P_{2} are not super vertex-graceful; for n ≥ 3, P_{n}^{2} × P_{2} is super vertex-graceful if and only if n = 3, 4, or 5; and C_{m} × C_{n} is not super vertex-graceful. They conjecture that P_{n} × P_{n} is super vertex-graceful for n ≥ 3.

In [591] Lee and Wong generalize super edge-vertex graphs by defining a graph G(V, E) to be P (a)Q(1)-super vertex-graceful if there is a bijection f from V to {0, ±a, ±(a + 1), …, ±(a − 1 + (|V | − 1)/2)} when |V | is odd and from V to {±a, ±(a + 1), …, ±(a − 1 + |V |/2)} when |V | is even such that the induced edge labeling f^{∗} defined by f^{+}(uv) = f(u)+f(v) over all edges uv is a bijection from E to {0, ±1, ±2, …, ±(|E|−1)/2} when |E| is odd and from E to {±1, ±2, …, ±|E|/2} when |E| is even. They show various classes of unicyclic graphs are P (a)Q(1)-super vertex-graceful. In [541] Lee, Leung, and Ng more simply refer to P (1)Q(1)-super vertex-graceful graphs as super vertex-graceful and show how to construct a variety of unicylic graphs that are super vertex-graceful. They conjecture that every unicylic graph is an induced subgraph of a super vertex-graceful unicyclic graph.

In [253] Chopra and Lee define a graph G(V, E) to be Q(a)P (b)-super edge-graceful if there is a bijection f from E to {±a, ±(a+1), …, ±(a+(|E|−2)/2)} when |E| is even and from E to {0, ±a, ±(a + 1), …, ±(a + (|E| − 3)/2)} when |E| is odd and f^{+}(u) is equal to the sum of f(uv) over all edges uv is a bijection from V to {±b, ±(b + 1), …, (|V | − 2)/2} when |V | is even and from V to {0, ±b, ±(b + 1), …, ±(|V | − 3)/2} when |V | is odd. They say a graph is strongly super edge-graceful if it is Q(a)P (b)-super edge-graceful for all a ≥ 1. Among their results are: a star with n pendent edges is strongly super edgegraceful if and only if n is even; wheels with n spokes are strongly super edge-graceful if and only if n is even; coronas C_{n} ⊙ K_{1} are strongly super edge-graceful for all n ≥ 3; and double stars DS(m, n) are strongly super edge-graceful in the case that m is odd and at least 3 and n is even and at least 2 and in the case that both m and n are odd and one of them is at least 3. Lee, Song, and Valdés [574] investigate the Q(a)P (b)-super edge-gracefulness of wheels W_{n} for n = 3, 4, 5, and 6.

In [588] Lee, Wang, and Yera proved that some Eulerian graphs are super edgegraceful, but not edge-graceful, and that some are edge-graceful, but not super edgegraceful. They also showed that a Rosa-type condition for Eulerian super edge-graceful graphs does not exist and pose some conjectures.

In 1997 Yilmaz and Cahit [992] introduced a weaker version of edge-graceful called E-cordial. Let G be a graph with vertex set V and edge set E and let f a function from E to {0, 1}. Define f on V by f(v) = ∑{f(uv)|uv ∈ E} (mod 2). The function f is called an E-cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1 and the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1. A graph that admits an E-cordial labeling is called E-cordial. Yilmaz and Cahit prove the following graphs are E-cordial: trees with n vertices if and only if n ≢ 2 (mod 4); K_{n} if and only if n ≢ 2 (mod 4); K_{m,n} if and only if m + n ≢ 2 (mod 4); C_{n} if and only if n ≢ 2 (mod 4); regular graphs of degree 1 on 2n vertices if and only if n is even; friendship graphs C_{3}^{(n)} for all n (see §2.2 for the definition); fans F_{n} if and only if n ≢ 1 (mod 4); and wheels W_{n} if and only if n ≢ 1 (mod 4). They observe that graphs with n ≡ 2 (mod 4) vertices can not be E-cordial. They generalize E-cordial labelings to E_{k}-cordial (k > 1) labelings by replacing {0, 1} by {0, 1, 2, …, k − 1}. Of course, E_{2}-cordial is the same as E-cordial.

Devaraj [280] has shown that M(m, n), the mirror graph of K(m, n) (see §2.3 for the definition), is E-cordial when m + n is even and the generalized Petersen graph P (n, k) is E-cordial when n is even. (Recall that P (n, 1) is C_{n} × P_{2}.)

The table following summarizes the state of knowledge about edge-graceful labelings. In the table EG means edge-graceful labeling exists. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property.

Table 22: Summary of Edge-graceful Labelings

Graph  Types Notes
K_{n}  EG iff n ≢ 2 (mod 4) [530]
odd order trees  EG? [532]
K_{n,n,...,n} (k terms)  EG iff n is odd or k ≢ 2 (mod 4) [565]
C_{n}^{k}, k < ⌊n/2⌋  EG iff n is odd [564]
C_{n}^{k}, k ≥ ⌊n/2⌋  EG iff n ≢ 2 (mod 4) [564]
P_{3}[K_{n}]  EG n is odd [564]
M_{4n} (Möbius ladders)  EG [533]
odd order dragons  EG [520]
odd order unicycilc graphs  EG? [520]
P_{2m} × P_{2n}  EG iff m = n = 2 [546]
C_{n} ∪ P_{2}  EG n even [568]
C_{2n} ∪ C_{2n+1}  EG n odd [568]
C_{n} ∪ C_{2n+2}  EG [568]
C_{n} ∪ C_{4n}  EG n odd [568]
C_{2m} ∪ C_{2n+1}  EG? (m, n) ≠ (4, 3) odd [569]
P (n, k) generalized Petersen graph EG n even, k < n/2 [533]
C_{m} × C_{n}  EG? (m, n) ≠ (4, 3) [569]

7.4 Radio Labelings

In 2001 Chartrand, Erwin, Zhang, and Harary [234] were motivated by regulations for channel assignments of FM radio stations to introduce radio labelings of graphs. A radio labeling of a connected graph G is an injection c from the vertices of G to the natural numbers such that

d(u, v) + |c(u) − c(v)| ≥ 1 + diam(G)

for every two distinct vertices u and v of G. The radio number of c, rn(c), is the maximum number assigned to any vertex of G. The radio number of G, rn(G), is the minimum value of rn(c) taken over all radio labelings c of G. Chartrand et al. and Zhang [1008] gave bounds for the radio numbers of cycles. The exact values for the radio numbers for paths and cycles were reported by Liu and Zhu [612] as follows: for odd n ≥ 3, rn(P_{n}) = (n − 1)^{2}/2+2; for even n ≥ 4, rn(P_{n}) = n^{2}/2−n+1; rn(C_{4k}) = (k+2)(k−2)/2+1; rn(C_{4k+1}) = (k + 1)(k − 1)/2; rn(C_{4k+2}) = (k + 2)(k − 2)/2 + 1; and rn(C_{4k+3}) = (k + 2)(k − 1)/2. However, Chartrand, Erwin, and Zhang [233] obtained different values than Liu and Zhu for P_{4} and P_{5}. Chartrand, Erwin, and Zhang [233] proved: rn(P_{n}) ≤ (n − 1)(n − 2)/2 + n/2+1 when n is even; rn(P_{n}) ≤ n(n−1)/2+1 when n is odd; rn(P_{n}) < rn(P_{n+1}) (n > 1); for a connected graph G of diameter d, rn(G) ≥ (d + 1)^{2}/4 + 1 when d is odd; and rn(G) ≥ d(d + 2)/4 + 1 when d is even. In [609] Liu obtained lower bounds for the radio number of trees and the radio number of spiders (trees with at most one vertex of degree greater than 2) and characterized the graphs that achieve these bounds.

Chartrand, Erwin, Zhang, and Harary [234] proved: rn(K_{n_{1},n_{2},...,n_{k}}) = n_{1} + n_{2} + ⋯ + n_{k} + k − 1; if G is a connected graph of order n and diameter 2, then n ≤ rn(G) ≤ 2n − 2; and for every pair of integers k and n with n ≤ k ≤ 2n−2, there exists a connected graph of order n and diameter 2 with rn(G) = k. They further provide a characterization of connected graphs of order n and diameter 2 with prescribed radio number.

Liu and Xie [611] investigated the radio numbers of squares of cycles. Letting n = 4k+r where r = 0, 1, 2 or 3, they proved: rn(C_{n}^{2}) = (2k^{2} + 5k − 1)/2, if r = 0 and k is odd; rn(C_{n}^{2}) = (2k^{2} + 3k)/2, if r = 0 and k is even; rn(C_{n}^{2}) ≥ (k^{2} + k), if r = 1 and k ≡ 1 (mod 4); rn(C_{n}^{2}) = (k^{2} + k), if r = 1 and k ≡ 3 (mod 4); rn(C_{n}^{2}) = (k^{2} + 2k), if r = 1 and k is even; rn(C_{n}^{2}) = (k^{2} + 5k + 1), if r = 2 and k is odd; rn(C_{n}^{2}) = (k^{2} + 4k + 1), if r = 2 and k ≡ 2 (mod 4); rn(C_{n}^{2}) ≥ (2k^{2} + 7k + 3)/2, if r = 3 and k is odd; rn(C_{n}^{2}) ≥ (2k^{2} + 9k + 4)/2, if r = 3 and k is even; rn(C_{n}^{2}) = (2k^{2} + 5k − 1)/2, if r = 2 and k is odd; rn(C_{n}^{2}) = (2k^{2} + 9k + 4)/2, if r = 3 and k ≡ 0 (mod 4); rn(C_{n}^{2}) = (2k^{2} + 9k + 4)/2, if r = 3 and k = 4m + 2 for some m ≢ 5 (mod 7); rn(C_{n}^{2}) = (2k^{2} + 7k + 3)/2 if r = 3 and k = 4m + 1 where m ≡ 0 or 1 (mod 3); rn(C_{n}^{2}) = (2k^{2} + 7k + 5)/2 if r = 3 and k = 4m + 1 where m ≡ 2 (mod 3); rn(C_{n}^{2}) ≥ (2k^{2} + 7k + 5)/2 if r = 3 and k = 4m + 3 where m ≡ 0 (mod 3); rn(C_{n}^{2}) ≥ (2k^{2} + 7k + 3)/2 if r = 3 and k = 4m + 3 where m ≡ 1 or 2 (mod 3). They also prove that if r = 3 for some k = 4m + 2 and m ≡ 5 (mod 7), then (2k^{2} + 9k + 4)/2 ≤ rn(C_{n}^{2}) ≤ 2k^{2} + 9k + 10)/2 and conjecture that if k ≡ 3 (mod 4), then rn(C_{4k+1}^{2}) = k^{2} + k + 2.

In [610] Liu found a lower bound for the radio number of trees and characterizes the trees that achieve the bound. She also provides a lower bound for the radio number of spiders in terms of the lengths of their legs and characterizes the spiders that achieve this bound.

Sooryanarayana and Raghunath [869] say a graph with n vertices is radio graceful if rn(G) = n. They determine the radio number of the cube of C_{n} for all n ≤ 20 and for n ≡ 0 or 2 or 4 (mod 6). They also determine the values of n for which C_{n}^{3} is radio graceful.

7.5 Line-graceful Labelings

Gnanajothi [367] has defined a concept similar to edge-graceful. She calls a graph with n vertices line-graceful if it is possible to label its edges with 0, 1, 2, …, n such that when each vertex is assigned the sum modulo n of all the edge labels incident with that vertex the resulting vertex labels are 0, 1, …, n − 1. A necessary condition for the line-gracefulness of a graph is that its order is not congruent to 2 (mod 4). Among line-graceful graphs are (see [367, pp. 132–181]) P_{n} if and only if n ≢ 2 (mod 4); C_{n} if and only if n ≢ 2 (mod 4); K_{1,n} if and only if n ≢ 1 (mod 4); P_{n} ⊙K_{1} (combs) if and only if n is even; (P_{n} ⊙K_{1})⊙K_{1} if and only if n ≢ 2 (mod 4); (in general, if G has order n, G ⊙ H is the graph obtained by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H); mC_{n} when mn is odd; C_{n} ⊙ K_{1} (crowns) if and only if n is even; mC_{4} for all m; complete n-ary trees when n is even; K_{1,n} ∪ K_{1,n} if and only if n is odd; odd cycles with a chord; even cycles with a tail; even cycles with a tail of length 1 and a chord; graphs consisting of two triangles having a common vertex and tails of equal length attached to a vertex other than the common one; the complete n-ary tree when n is even; trees for which exactly one vertex has even degree. She conjectures that all trees with p ≢ 2 (mod 4) vertices are line-graceful and proved this conjecture for p ≤ 9.

Gnanajothi [367] has investigated the line-gracefulness of several graphs obtained from stars. In particular, the graph obtained from K_{1,4} by subdividing one spoke to form a path of even order (counting the center of the star) is line-graceful; the graph obtained from a star by inserting one vertex in a single spoke is line-graceful if and only if the star has p ≢ 2 (mod 4) vertices; the graph obtained from K_{1,n} by replacing each spoke with a path of length m (counting the center vertex) is line-graceful in the following cases: n = 2; n = 3 and m ≢ 3 (mod 4); and m is even and mn + 1 ≡ 0 (mod 4).

Gnanajothi studied graphs obtained by joining disjoint graphs G and H with an edge. She proved such graphs are line-graceful in the following circumstances: G = H; G = P_{n}, H = P_{m} and m + n ≢ 0 (mod 4); and G = P_{n} ⊙ K_{1}, H = P_{m} ⊙ K_{1} and m + n ≢ 0 (mod 4).

7.6 Representations of Graphs modulo n

In 1989 Erdős and Evans [307] defined a representation modulo n of a graph G with vertices v_{1}, v_{2}, …, v_{r} as a set {a_{1}, …, a_{r}} of distinct, nonnegative integers each less than n satisfying gcd(a_{i} − a_{j}, n) = 1 if and only if v_{i} is adjacent to v_{j}. They proved that every finite graph can be represented modulo some positive integer. The representation number, Rep(G), is smallest such integer. Obviously the representation number of a graph is prime if and only if a graph is complete. Evans, Fricke, Maneri, McKee, and Perkel [316] have shown that a graph is representable modulo a product of a pair of distinct primes if and only if the graph does not contain an induced subgraph isomorphic to K_{2} ∪2K_{1}, K_{3} ∪K_{1}, or the complement of a chordless cycle of length at least five. Nešetřil and Pultr [679] showed that every graph can be represented modulo a product of some set of distinct primes. Evans et al. [316] proved that if G is representable modulo n and p is a prime divisor of n, then p ≥ χ(G). Evans, Isaak, and Narayan [317] determined representation numbers for specific families as follows (here we use q_{i} to denote the ith prime and for any prime p_{i} we use p_{i+1}, p_{i+2}, …, p_{i+k} to denote the next k primes larger than p_{i}): Rep(P_{n}) = 2·3·· · ··q_{⌈log_{2}(n−1)⌉}; Rep(C_{4}) = 4 and for n ≥ 3, Rep(C_{2n}) = 2·3·· · ··q_{⌈log_{2}(n−1)⌉+1}; Rep(C_{5}) = 3 · 5 · 7 = 105 and for n ≥ 4 and not a power of 2, Rep(C_{2n+1}) = 3 · 5 · · · · · q_{⌈log_{2}n⌉+1}; if m ≥ n ≥ 3, then Rep(K_{m} − P_{n}) = p_{i}p_{i+1} where p_{i} is the smallest prime greater than or equal to m − n + ⌈n/2⌉; if m ≥ n ≥ 4, and p_{i} is the smallest prime greater than or equal to m − n + ⌈n/2⌉, then Rep(K_{m} − C_{n}) = q_{i}q_{i+1} if n is even and Rep(K_{m} − C_{n}) = q_{i}q_{i+1}q_{i+2} if n is odd; if n ≤ m − 1, then Rep(K_{m} − K_{1,n}) = p_{s}p_{s+1} ⋯ p_{s+n−1} where p_{s} is the smallest prime greater than or equal to m − 1; Rep(K_{m}) is the smallest prime greater than or equal to m; Rep(nK_{2}) = 2 · 3 · · · · · q_{⌈log_{2}n⌉+1}; if n, m ≥ 2, then Rep(nK_{m}) = p_{i}p_{i+1} ⋯ p_{i+m−1}, where p_{i} is the smallest prime satisfying p_{i} ≥ m, if and only if there exists a set of n − 1 mutually orthogonal Latin squares of order m; Rep(mK_{1}) = 2m; and if t ≤ (m − 1)!, then Rep(K_{m} + tK_{1}) = p_{s}p_{s+1} ⋯ p_{s+m−1} where p_{s} is the smallest prime greater than or equal to m. Narayan [678] proved that for r ≥ 3 the maximum value for Rep(G) over all graphs of order r is p_{s}p_{s+1} ⋯ p_{s+r−2}, where p_{s} is the smallest prime that is greater than or equal to r − 1.

Evans [315] used matrices over the additive group of a finite field to obtain various bounds for the representation number of graphs of the form nK_{m}. Among them are Rep(4K_{3}) = 3 · 5 · 7 · 11; Rep(7K_{5}) = 5 · 7 · 11 · 13 · 17 · 19 · 23; and Rep((3q − 1)/2)K_{q}) ≤ p_{q}p_{q+1} ⋯ p_{(3q−1)/2)} where q is a prime power with q ≡ 3 (mod 4), p_{q} is the smallest prime greater than or equal to q, and the remaining terms are the next consecutive (3q − 3)/2 primes; Rep(2q − 2)K_{q}) ≤ p_{q}p_{q+1} ⋯ p_{(3q−3)/2)} where q is a prime power with q ≡ 3 mod 4, and p_{q} is the smallest prime greater than or equal to q; Rep((2q−2)K_{q}) ≤ p_{q}p_{q+1} ⋯ p_{2q−3}.

In [677] Narayan asked for the values of Rep(C_{2^{k}+1}) when k ≥ 3 and Rep(G) when G is a complete multipartite graph or a disjoint union of complete graphs. He also asked about the behavior of the representation number for random graphs.

7.7 k-sequential Labelings

In 1981 Bange, Barkauskas, and Slater [127] defined a k-sequential labeling f of a graph G(V, E) as one for which f is a bijection from V ∪E to {k, k +1, …, |V ∪E|+k −1} such that for each edge xy in E, f(xy) = |f(x) − f(y)|. This generalized the notion of simply sequential where k = 1 introduced by Slater. Bange, Barkauskas, and Slater showed that cycles are 1-sequential and if G is 1-sequential, then G+K_{1} is graceful. Hegde and Shetty [424] have shown that every T_{p}-tree (see §4.4 for the definition) is 1-sequential. In [849], Slater proved: K_{n} is 1-sequential if and only if n ≤ 3; for n ≥ 2, K_{n} is not k-sequential for any k ≥ 2; and K_{1,n} is k-sequential if and only if k divides n. Acharya and Hegde [21] proved: if G is k-sequential, then k is at most the independence number of G; P_{2n} is nsequential for all n and P_{2n+1} is both n-sequential and (n + 1)-sequential for all n; K_{m,n} is k-sequential for k = 1, m, and n; K_{m,n,1} is 1-sequential; and the join of any caterpillar and \overline{K_{t}} is 1-sequential. Acharya [12] showed that if G(E, V) is an odd graph with |E|+|V | ≡ 1 or 2 (mod 4) when k is odd or |E| + |V | ≡ 2 or 3 (mod 4) when k is even, then G is not k-sequential. Acharya also observed that as a consequence of results of Bermond, Kotzig, and Turgeon [162] we have: mK_{4} is not k-sequential for any k when m is odd and mK_{2} is not k-sequential for any odd k when m ≡ 2 or 3 (mod 4) or for any even k when m ≡ 1 or 2 (mod 4). He further noted that K_{m,n} is not k-sequential when k is even and m and n are odd, whereas K_{m,k} is k-sequential for all k. Acharya [12] points out that the following result of Slater’s [850] for k = 1 linking k-graceful graphs and k-sequential graphs holds in general: A graph is k-sequential if and only if G + v has a k-graceful labeling f with f(v) = 0. Slater [849] also proved that a k-sequential graph with p vertices and q > 0 edges must satisfy k ≤ p−1. Hegde [415] proved that every graph can be embedded as an induced subgraph of a simply sequential graph. In [12] Acharya conjectured that if G is a connected k-sequential graph of order p with k > ⌊p/2⌋, then k = p − 1 and G = K_{1,p−1} and that, except for K_{1,p−1}, every tree in which all vertices are odd is k-sequential for all odd positive integers k ≤ p/2. In [415] Hegde gave counterexamples for both of these conjectures.

In [423] Hegde and Miller prove the following: for n > 1, K_{n} is k-sequentially additive if and only if (n, k) = (2, 1), (3, 1) or (3,2); K_{1,n} is k-sequentially additive if and only if k divides n; caterpillars with bipartition sets of sizes m and n are k-sequentially additive for k = m and k = n; and if an odd-degree (p, q)-graph is k-sequentially additive, then (p + q)(2k + p + q − 1) ≡ 0 mod 4. As corollaries of the last result they observe that when m and n are odd and k is even K_{m,n} is not k-sequentially additive and if an odd-degree tree is k-sequentially additive then k is odd.

7.8 IC-colorings

For a subgraph H of a graph G with vertex set V and a coloring f from V to the natural numbers define f_{s}(H) = Σf(v) over all v ∈ H. The coloring f is called an IC-coloring if for any integer k between 1 and f_{s}(G) there is a connected subgraph H of G such that f_{s}(H) = k. The IC-index of a graph G, M(G), is max{f_{s}| f_{s} is an IC-coloring of G}. Salehi, Lee, and Khatirinejad [741] obtained the following: M(K_{n}) = 2^{n} − 1; for n ≥ 2, M(K_{1,n}) = 2^{n} + 2; if Δ is the maximum degree of a connected graph G, then M(G) ≥ 2^{Δ} + 2; if ST (n; 3^{n}) is the graph obtained by identifying the end points of n paths of length 3, then ST (n; 3^{n}) is at least 3^{n} + 3 (they conjecture that equality holds for n ≥ 4); for n ≥ 2, M(K_{2,n}) = 3 · 2^{n} + 1; M(P_{n}) ≥ (2 + ⌊n/2⌋)(n − ⌊n/2⌋) + ⌊n/2⌋ − 1; for m, n ≥ 2, the IC-index of the double star DS(m, n) is at least (2^{m−1} + 1)(2^{n−1} + 1) (they conjecture that equality holds); for n ≥ 3, n(n + 1)/2 ≤ M(C_{n}) ≤ n(n − 1) + 1; and for n ≥ 3, 2^{n} + 2 ≤ M(W_{n}) ≤ 2^{n} + n(n − 1) + 1. They pose the following open problems: find the IC-index of the graph obtained by identifying the end points of n paths of length b; find the IC-index of the graph obtained by identifying the end points of n paths; and find the IC-index of K_{m,n}. Shiue and Fu [828] completed the partial results by Penrice [699] Salehi, Lee, and Khatirinejad [741] by proving M(K_{m,n}) = 3 · 2^{m+n−2} − 2^{m−2} + 2 for any 2 ≤ m ≤ n.

7.9 Product Cordial Labelings

Sundaram and Somasundaram [894] introduced the notion of product cordial labelings. A product cordial labeling of a graph G with vertex set V is a function f from V to {0, 1} such that if each edge uv is assigned the label f(u)f(v), the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1.

A graph with a product cordial labeling is called a product cordial graph. In [889] and [893] Sundaram, Ponraj, and Somasundaram prove the following graphs are product cordial: trees; unicyclic graphs of odd order; triangular snakes; dragons; helms; P_{m} ∪ P_{n}; C_{m} ∪P_{n}; P_{m} ∪K_{1,n}; W_{m} ∪F_{n} (F_{n} is the fan P_{n} +K_{1}); K_{1,m} ∪K_{1,n}; W_{m} ∪K_{1,n}; W_{m} ∪ P_{n}; W_{m} ∪ C_{n}; the total graph of P_{n} (the total graph of P_{n} has vertex set V (P_{n}) ∪ E(P_{n}) with two vertices adjacent whenever they are neighbors in P_{n}); C_{n} if and only if n < 4; C_{n}^{(t)}, the one-point union of t copies of C_{n}, provided t is even or both t and n are even; K_{2} + mK_{1} if and only if m is odd; C_{m} ∪ P_{n} if and only if m + n is odd; K_{m,n} ∪ P_{s} if s > mn; C_{n+2} ∪ K_{1,n}; K_{n} ∪ K_{n,(n−1)/2} when n is odd; K_{n} ∪ K_{n−1,n/2} when n is even; and P_{n}^{2} if and only if n is odd. They also prove that K_{m,n} (m, n > 2), P_{m} × P_{n} (m, n > 2) and wheels are not product cordial and if a (p, q)-graph is product cordial graph, then q < (p − 1)(p + 1)/4.

Sundaram and Somasundaram [894] also have introduced the notion of total product cordial labelings. A total product cordial labeling of a graph G with vertex set V is a function f from V to {0, 1} such that if each edge uv is assigned the label f(u)f(v) the number of vertices and edges labeled with 0 and the number of vertices and edges labeled with 1 differ by at most 1. A graph with a total product cordial labeling is called a total product cordial graph. In [894] and [891] Sundaram, Ponraj, and Somasundaram prove the following graphs are total product cordial: every product cordial graph of even order or odd order and even size; trees; all cycles except C_{4}; K_{n,2n−1}; C_{n} with m edges appended at each vertex; fans; double fans; wheels; helms; C_{2} × P_{2}; K_{2,n} if and only if n ≡ 2 (mod 4); P_{m} × P_{n} if and only if (m, n) ≠ (2, 2); C_{n} + 2K_{1} if and only if n is even or n ≡ 1 (mod 3); \overline{K_{n}} × 2K_{2} if n is odd, or n ≡ 0 or 2 (mod 6), or n ≡ 2 (mod 8).

7.10 Prime Cordial Labelings

Sundaram, Ponraj, and Somasundaram [890] have introduced the notion of prime cordial labelings. A prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, …, |V |} such that if each edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and 0 if gcd(f(u), f(v)) > 1, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In [890] Sundaram, Ponraj, and Somasundram prove the following graphs are prime cordial: C_{n} if and only if n ≥ 6; P_{n} if and only if n ≠ 3 or 5; K_{1,n} (n odd); the graph obtained by subdividing each edge of K_{1,n} if and only if n ≥ 3; bistars; dragons; crowns; triangular snakes if and only if the snake has at least three triangles; ladders; K_{1,n} if n is even and there exists a prime p such that 2p < n + 1 < 3p; K_{2,n} if n is even and if there exists a prime p such that 3p < n + 2 < 4p; and K_{3,n} if n is odd and if there exists a prime p such that 5p < n + 3 < 6p. They also prove that if G is a prime cordial graph of even size, then the graph obtained by identifying the central vertex of K_{1,n} with the vertex of G labeled with 2 is prime cordial, and if G is a prime cordial graph of odd size, then the graph obtained by identifying the central vertex of K_{1,2n} with the vertex of G labeled with 2 is prime cordial. They further prove that K_{n} is not prime cordial for 4 < n < 181 and K_{m,n} is not prime cordial for a number of special cases of m and n.

7.11 Geometric Labelings

If a and r are positive integers at least 2, we say a (p, q)-graph G is (a, r)-geometric if its vertices can be assigned distinct positive integers such that the value of the edges obtained as the product of the end points of each edge is {a, ar, ar^{2}, …, ar^{q−1}}. Hegde [418] has shown the following: no connected bipartite graph, except the star, is (a, a)-geometric where a is a prime number or square of a prime number; any connected (a, a)-geometric graph where a is a prime number or square of a prime number, is either a star or has a triangle; K_{a,b}, 2 ≤ a ≤ b is (k, k)-geometric if and only if k is neither a prime number nor the square of a prime number; a caterpillar is (k, k)-geometric if and only if k is neither a prime number nor the square of a prime number; K_{a,b,1} is (k, k)-geometric for all integers k ≥ 2; C_{4t} is (a, a)-geometric if and only if a is neither a prime number nor the square of a prime number; for any positive integers t and r ≥ 2, C_{4t+1} is (r^{2t}, r)-geometric; for any positive integer t, C_{4t+2} is not geometric for any values of a and r; and for any positive integers t and r ≥ 2, C_{4t+3} is (r^{2t+1}, r)-geometric. Hegde [420] has also shown that every T_{p}-tree and the subdivision graph of every T_{p}-tree are (a, r)-geometric for some values of a and r (see Section 3.2 for the definition of a T_{p}-tree). He conjectures that all trees are (a, r)-geometric for some values of a and r.

Hegde and Shankaran [425] prove: a graph with an α-labeling (see §3.1 for the definition) where m is the fixed integer that is between the end points of each edge has an (a^{m+1}, a)-geometric for any a > 1; for any integers m and n both greater than 1 and m odd, mP_{n} is (a^{r}, a)-geometric where r = (mn + 3)/2 if n is odd and (a^{r}, a)-geometric where r = (m(n + 1) + 3)/2 if n is even; for positive integers k > 1, d ≥ 1, and odd n, the generalized closed helm (see §5.3 for the definition) CH(t, n) is (k^{r}, k^{d})-geometric where r = (n − 1)d/2; for positive integers k > 1, d ≥ 1, and odd n, the generalized web graph (see §5.3 for the definition) W (t, n) is (k^{r}, a)-geometric where a = k^{d} and r = (n − 1)d/2; for positive integers k > 1, d ≥ 1, the generalized n-crown (P_{m} × K_{3}) ⊙ K_{1,n} is (a, a)- geometric where a = k^{d}; and n = 2r + 1, C_{n} ⊙ P_{3} is (k^{r}, k)-geometric.

7.12 Sequentially Additive Graphs

Bange, Barkauskas, and Slater [128] defined a k-sequentially additive labeling f of a graph G(V, E) to be a bijection from V ∪ E to {k, …, k + |V ∪ E| − 1} such that for each edge xy, f(xy) = f(x) + f(y). They proved: K_{n} is 1-sequentially additive if and only if n ≤ 3; C_{3n+1} is not k-sequentially additive for k ≡ 0 or 2 (mod 3); C_{3n+2} is not ksequentially additive for k ≡ 1 or 2 (mod 3); C_{n} is 1-sequentially additive if and only if n ≡ 0 or 1 (mod 3); and P_{n} is 1-sequentially additive. They conjecture that all trees are 1-sequentially additive. Hegde [417] proved that K_{1,n} is k-sequentially additive if and only if k divides n.

Acharya and Hegde [23] have generalized k-sequentially additive labelings by allowing the image of the bijection to be {k, k +d, …, (k +|V ∪E|−1)d}. They call such a labeling additively (k, d)-sequential.

7.13 Strongly Multiplicative Graphs

Beineke and Hegde [153] call a graph with p vertices strongly multiplicative if the vertices of G can be labeled with distinct integers 1, 2, …, p such that the labels induced on the edges by the product of the end vertices are distinct. They prove the following graphs are strongly multiplicative: trees; cycles; wheels; K_{n} if and only if n ≤ 5; K_{r,r} if and only if r ≤ 4; and P_{m} × P_{n}. They then consider the maximum number of edges a strongly multiplicative graph on n vertices can have. Denoting this number by λ(n), they show: λ(4r) ≤ 6r^{2}; λ(4r + 1) ≤ 6r^{2} + 4r; λ(4r + 2) ≤ 6r^{2} + 6r + 1; and λ(4r + 3) ≤ 6r^{2} + 10r + 3. Adiga, Ramaswamy, and Somashekara [33] give the bound λ(n) ≤ n(n + 1)/2 + n − 2 − ⌊(n + 2)/4⌋ − ∑_{i=2}^{n} i/p(i) where p(i) is the smallest prime dividing i. For large values of n this is a better upper bound for λ(n) than the one given by Beineke and Hegde. It remains an open problem to find a nontrivial lower bound for λ(n).

Seoud and Zid [774] prove the following graphs are strongly multiplicative: wheels; rK_{n} for all r and n at most 5; rK_{n} for r ≥ 2 and n = 6 or 7; rK_{n} for r ≥ 3 and n = 8 or 9; K_{4,r} for all r; and the corona of P_{n} and K_{m} for all n and 2 ≤ m ≤ 8.

Germina and Ajitha [362] (see also [17]) prove that K_{2} + \overline{K_{t}}, quadrilateral snakes, Petersen graphs, ladders, and unicyclic graphs are strongly multiplicative. Acharya, Germina, and Ajitha [17] have shown that C_{k}^{(n)} (see §2.2 for the definition) is strongly multiplicative and that every graph can be embedded as an induced subgraph of a strongly multiplicative graph. Germina and Ajitha [362] define a graph with q edges and a strongly multiplicative labeling to be hyper strongly multiplicative if the induced edge labels are {2, 3, …, q + 1}. They show that every hyper strongly multiplicative graph has exactly one nontrivial component that is either a star or has a triangle and every graph can be embedded as an induced subgraph of a hyper strongly multiplicative graph.

7.14 Mean Labelings

Somasundaram and Ponraj [861] have introduced the notion of mean labelings of graphs. A graph G with p vertices and q edges is called a mean graph if there is an injective function f from the vertices of G to {0, 1, 2, …, q} such that when each edge uv is labeled with (f(u) + f(v))/2 if f(u) + f(v) is even, and (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd, then the resulting edge labels are distinct. In [861], [862], [863], [864], and [709] they prove the following graphs are mean graphs: P_{n}, C_{n}, K_{2,n}, K_{2} + mK_{1}, \overline{K_{n}} + 2K_{2}, C_{m} ∪ P_{n}, P_{m} × P_{n}, P_{m} × C_{n}, C_{m} ⊙ K_{1}, P_{m} ⊙ K_{1}, triangular snakes, quadrilateral snakes, K_{n} if and only if n < 3, K_{1,n} if and only if n < 3, bistars B_{m,n} (m > n) if and only if m < n+2, the subdivision graph of the star K_{1,n} if and only if n < 4, and the friendship graph C_{3}^{(t)} if and only if t < 2. They also prove that W_{n} is not a mean graph for n > 3 and enumerate all mean graphs of order less than 5.

Gayathri and Amuthavalli [358] say a (p, q)-graph G has a (k, d)-odd mean labeling if there exists an injection f from the vertices of G to {0, 1, 2, …, 2k − 1 + 2(q − 1)d} such that the induced map f^{∗} defined on the edges of G by f^{∗}(uv) = ⌈(f(u) + f(v))/2⌉ is a bijection from edges of G to {2k − 1, 2k − 1 + 2d, 2k − 1 + 4d, …, 2k − 1 + 2(q − 1)d}. When d = 1 a (k, d)-odd mean labeling is called k-odd mean . For n ≥ 2 they prove the following graphs are k-odd mean for all k : P_{n}; combs P_{n} ⊙ K_{1}; crowns C_{n} ⊙ K_{1} (n ≥ 4); bistars B_{n,n}; P_{m} ⊙ \overline{K_{n}} (m ≥ 2); C_{m} ⊙ \overline{K_{n}}; K_{2,n}; C_{n} except for n = 3 or 6; the one-point union of C_{n} (n ≥ 4) and an end point of any path; grids P_{m} ×P_{n} (m ≥ 2); (P_{n} ×P_{2})⊙K_{1}; arbitrary unions of paths; arbitrary unions of stars; arbitrary unions of cycles; the graphs obtained by joining two copies of C_{n} (n ≥ 4) by any path; and the graph obtained from P_{m} × P_{n} by replacing each edge by a path of length 2. They prove the following graphs are not k-odd mean for any k: K_{n}; K_{n} with an edge deleted; K_{3,n} (n ≥ 3); wheels; fans; friendship graphs; triangular snakes; Möbius ladders; books K_{1,m} ×P_{2} (m ≥ 4); and webs. For n ≥ 3 they prove K_{1,n} is k-odd mean if and only if k ≥ n − 1. For n ≥ 2 the following graphs have a (k, d)-mean labeling [361]: C_{m} ∪ P_{n} (m ≥ 4) for all k; arbitrary unions of cycles for all k; P_{2m}; P_{2m+1} for k ≥ d (P_{2m+1} is not (k, d)-mean when k < d); combs P_{n} ⊙ K_{1} for all k; K_{1,n} for k ≥ d; K_{2,n} for k ≥ d; bistars for all k; nC_{4} for all k; and quadrilateral snakes for k ≥ d.

Gayathri and Tamilselvi [361] say a (p, q)-graph G has a (k, d)-super mean labeling if there exists an injection f from the vertices of G to {k, k + 1, …, p + q + k − 1} such that the induced map f^{∗} defined on the edges of G by f^{∗}(uv) = ⌈(f(u) + f(v))/2⌉ has the property that the vertex labels and the edge labels together are the integers from k to p + q + k − 1. When d = 1 a (k, d)-super mean labeling is called k-super mean. For n ≥ 2 they prove the following graphs are k-super mean for all k: odd cycles; P_{n}; C_{m} ∪ P_{n}; the one-point union of a cycle and the end point of P_{n}; the union of any two cycles excluding C_{4}; and triangular snakes. For n ≥ 2 they prove the following graphs are (k, d)-super mean for all k and d: P_{n}; odd cycles; combs P_{n} ⊙ K_{1}; and bistars.

Gayathri and Tamilselvi [361] say a (p, q)-graph G has a k-super edge mean labeling if there exists an injection f from the edges of G to {k, k + 1, …, k + 2(p + q)} such that the induced map f^{∗} from the vertices of G to {k, k + 1, …, k + 2(p + q)} defined by f^{∗}(v) = ⌈(Σf(vu))/2⌉ taken all edges vu incident to v is an injection. For n ≥ 3 they prove the following graphs are k-super edge mean for all k: paths; cycles; combs P_{n} ⊙ K_{1}; triangular snakes; crowns C_{n} ⊙ K_{1}; the one-point union of C_{3} and an end point of P_{n}; and P_{n} ⊙ K_{2}.

7.15 Permutation and Combination Graphs

Hegde and Shetty [430] define a graph G with p vertices to be a permutation graph if there exists a injection f from the vertices of G to {1, 2, 3, …, p} such that the induced edge function g_{f} defined by g_{f}(uv) = f(u)!/|f(u)−f(v)|! is injective. They say a graph G with p vertices is a combination graph if there exists a injection f from the vertices of G to {1, 2, 3, …, p} such that the induced edge function g_{f} defined as g_{f}(uv) = f(u)!/|f(u) − f(v)|!f(v)! is injective. They prove: K_{n} is a permutation graph if and only if n ≤ 5; K_{n} is a combination graph if and only if n ≤ 5; C_{n} is a combination graph for n > 3; K_{n,n} is a combination graph if and only if n ≤ 2; W_{n} is a not a combination graph for n ≤ 6; and a necessary condition for a (p, q)-graph to be a combination graph is that 4q ≤ p^{2} if p is even and 4q ≤ p^{2} − 1 if p is odd. They strongly believe that W_{n} is a combination graph for n ≥ 7 and all trees are combinations graphs. Baskar Babujee and Vishnupriya [117] prove the following graphs are permutation graphs: P_{n}; C_{n}; stars; graphs obtained adding a pendent edge to each edge of a star; graphs obtained by joining the centers of two identical stars with an edge or a path of length 2); and complete binary trees with at least three vertices.

7.16 Strongly ⋆-graphs

A variation of strong multiplicity of graphs is a strongly ⋆-graph. A graph of order n is said to be a strongly ⋆-graph if its vertices can be assigned the values 1, 2, …, n in such a way that, when an edge whose vertices are labeled i and j is labeled with the value i + j + ij, all edges have different labels. Adiga and Somashekara [34] have shown that all trees, cycles, and grids are strongly ⋆-graphs. They further consider the problem of determining the maximum number of edges in any strongly ⋆-graph of given order and relate it to the corresponding problem for strongly multiplicative graphs.

Baskar Babujee and Vishnupriya [117] have proved the following are strongly ⋆-graphs: C_{n} × P_{2}, (P_{2} ∪ \overline{K}_{m}) + \overline{K}_{2}, windmills K_{3}^{(n)}, and jelly fish graphs jelly fish J(m, n) obtained from a 4-cycle v_{1}, v_{2}, v_{3}, v_{4} by joining v_{1} and v_{3} with an edge and appending m pendent edges to v_{2} and n pendent edges to v_{4}.

7.17 Irregular total Labelings

Motivated by the notion of the irregularity strength of a graph introduced by Chartrand, Jacobson, Lehel, Oellermann, Ruiz, and Saba [235] in 1988 and various kinds of other total labelings, Bača, Jendroľ, Miller, and Ryan [93] introduced the total edge irregularity strength of a graph as follows. For a graph G(V, E) a labeling ∂ : V ∪ E → {1, 2, …, k} is called an edge irregular total k-labeling if for every pair of distinct edges uv and xy, ∂(u)+ ∂(uv)+∂(v) ≠ ∂(x)+∂(xy)+∂(y). Similarly, ∂ is called an vertex irregular total k-labeling if for every pair of distinct vertices u and v, ∂(u)+∑ ∂(e) over all edges e incident to u ≠ ∂(v) + ∑ ∂(e) over all edges e incident to v. The minimum k for which G has an edge (vertex) irregular total k-labeling is called the total edge (vertex) irregularity strength of G. The total edge (vertex) irregular strength of G is denoted by tes(G) (tvs(G)). They prove: for G(V, E), E not empty, ⌈(|E| + 2)/3⌉ ≤ tes(G) ≤ |E|; tes(G) ≥ ⌈(Δ(G) + 1)/2⌉ and tes(G) ≤ |E| − Δ(G), if Δ(G) ≤ (|E| − 1)/2; tes(P_{n}) = tes(C_{n}) = ⌈(n + 2)/3⌉; tes(W_{n}) = ⌈(2n + 2)/3⌉; tes(C_{3}^{n)} (friendship graph)= ⌈(3n + 2)/3⌉; tvs(C_{n}) = ⌈(n + 2)/3⌉; for n ≥ 2, tvs(K_{n}) = 2; tvs(K_{1,n}) = ⌈(n+1)/2⌉; and tvs(C_{n}×P_{2}) = ⌈(2n+3)/4⌉. Jendrol’, Miškul, and Soták [457] proved: tes(K_{5}) = 5; for n ≥ 6, tes(K_{n}) = ⌈(n^{2} − n + 4)/6⌉; and that tes(K_{m,n}) = ⌈(mn + 2)/3⌉. They conjecture that for any graph G other than K_{5}, tes(G) = max{⌈(Δ(G) + 1)/2⌉, ⌈(|E| + 2)/3⌉}. Ivančo and Jendrol’[457] proved that this conjecture is true for all trees. Brandt, Misškuf, and Rautenbach [192] proved the conjecture for large graphs whose maximum degree is not too large relative to its order and size. In particular, using the probabilistic method they prove that if G(V, E) is a multigraph without loops and with nonzero maximum degree less than |E|/10^{3}\sqrt{8|V |}, then tes(G) = (⌈|E|+2)/3⌉. As corollaries they have: if G(V, E) satisfies |E| ≥ 3·10^{3}n^{3/2}, then tes(G) = ⌈(|E| + 2)/3⌉; if G(V, E) has minimum degree δ > 0 and maximum degree Δ such that Δ < δ\sqrt{|V |}/10^{3} · 4\sqrt{2} then tes(G) = ⌈(|E| + 2)/3⌉; and for every positive integer Δ there is some n(Δ) such that every graph G(V, E) without isolated vertices with |V | ≥ n(Δ) and maximum degree at most Δ satisfies tes(G) = ⌈(|E| + 2)/3⌉. Notice that this last result includes d-regular graphs of large order. They also prove that if G(V, E) has maximum degree Δ ≥ 2|E|/3, then G has an edge irregular total k-labeling with k = ⌈(Δ + 1)/2⌉.

In [694] Nurdin, A.N.M. Salman, and E.T. Baskoro determine the total edge-irregular strengths of graphs of the form P_{n} ⊙ G (corona product) in the cases where G is a path, cycle, star, wheel, gear graph, or friendship graph. Ahmad and Bača [37] proved tvs(J_{n,2}) = ⌈(n + 1)/2)⌉ (n ≥ 4) and conjectured that for n ≥ 3 and m ≥ 3, tvs(J_{n,m}) = max{⌈(n(m − 1) + 2)/3⌉, ⌈(nm + 2)/4⌉}. They also proved that for the circulant graph (see §5.1 for the definition) C_{n}(1, 2), n ≥ 5, tvs(C_{n}(1, 2) = ⌈(n + 4)/5⌉. They conjecture that for the circulant graph C_{n}(a_{1}, a_{2}, …, a_{m}) with degree at least 5 and n ≥ 5, 1 ≤ a_{i} ≤ ⌊n/2⌋, tvs(C_{n}(a_{1}, a_{2}, …, a_{m}) = ⌈(n + r)/(1 + r)⌉.

7.18 Sigma Labelings

Vilfred and Jinnah [927] call a labeling f from V (G) to {1, 2, …, |V (G)|} a sigma labeling if for every vertex u the sum of all f(v) such that v is adjacent to u is a constant independent of u. This notion was first introduced by Vilfred in his Ph. D. thesis in 1994. In [927] Vilfred and Jinnah give a number of necessary conditions for a graph to have a sigma labeling. One of them is that if u and v are vertices of a graph with a sigma labeling, then the order of the symmetric difference of N(u) and N(v) (neighborhoods of u and v) is not 1 or 2. This condition rules out a large class of graphs as having sigma labelings. Vilfred and Jinnah raise a number of open questions: do there exist connected graphs that have sigma labelings other than complete multipartite graphs (in [926] it is shown that K_{2,2,...,2} has a sigma labeling); which complete multipartite graphs have sigma labelings; is it true that P_{m} × C_{n} (m > 1) does not have a sigma labeling; and is every graph an induced subgraph of a graph with a sigma labeling (they show that every graph is a subgraph of a graph with a sigma labeling).

The concept of sigma labeling was independently studied by Simanjuntak, Rodgers and Miller in [831] under the name 1-vertex magic vertex labeling. Among their results are: the only trees that have sigma labelings are P_{1} and P_{3}; C_{n} has a sigma labeling if and only if n = 4; K_{n} has a sigma labeling if and only if n = 1; the wheel W_{n} = C_{n} + P_{1} has a sigma labeling if and only if n = 4; the complete graph K_{n,n,...,n} with p partite sets has a sigma labeling if and only if n is even or both n and p are odd; and G × \overline{K_{2n}} has a sigma labeling for any regular graph G. Acharaya, Rao, Sighn, and Parameswaran [27] proved P_{n} × C_{k} does not have a sigma labeling when n and k are at least 3 and provide necessary and sufficient conditions for K_{n_{1},n_{2}} to have a sigma labeling. Rao, Singh, and Paraeswaran [721] have shown C_{n} × C_{k}, n ≥ 3, k ≥ 3 has a sigma labeling if and only if n = k ≡ 2 (mod 4) and K_{m} × K_{n}, m ≥ 2, n ≥ 3 does not have a sigma labeling. In [152] gives necessary and sufficient condition for K_{m,n} to be a sigma labeled graph and proves that if G_{1} and G_{2} are connected graphs with minimum degree 1 and at least three vertices, then G_{1} × G_{2} does not have a sigma labeling. This question was answered by Acharaya, Rao, Sighn, and Parameswara [28] who prove that every graph is an induced subgraph of a regular graph that has a sigma labeling. As open problems, Rao [720] asks for a characterize 4-regular graphs that have sigma labelings and which graphs of the form C_{m} × C_{n}, m = n ≡ 2 (mod 4) have sigma labelings.

7.19 Set Graceful and Set Sequential Graphs

The notions of set graceful and set sequential graphs were introduced by Acharaya in 1983 [13]. A graph is called set graceful if there is an assignment of nonempty subsets of a finite set to the vertices and edges of the graph such that the value given to each edge is the symmetric difference of the sets assigned to the endpoints of the edge, the assignment of sets to the vertices is injective, and the assignment to the edges is bijective. A graph is called set sequential if there is an assignment of nonempty subsets of a finite set to the vertices and edges of the graph such that the value given to each edge is the symmetric difference of the sets assigned to the endpoints of the edge and the the assignment of sets to the vertices and the edges is bijective. The following has been shown: P_{n} (n > 3) is not set graceful [419]; C_{n} is not set sequential [22]; C_{n} is set graceful if and only if n = 2^{m} − 1 [421]; K_{n} is set graceful if and only if n = 2, 3 or 6 [665]; K_{n} (n ≥ 2) is set sequential if and only if n = 2 or 5 [421]; K_{a,b} is set sequential if and only if (a + 1)(b + 1) is a positive power of 2 [421]; a necessary condition for K_{a,b,c} to be set sequential is that a, b, and c cannot have the same parity [419]; K_{1,b,c} is not set sequential when b and c even [421]; K_{2,b,c} is not set sequential when b and c are odd [419]; no theta graph is set graceful [419]; the complete nontrivial n-ary tree is set sequential if and only if n + 1 is a power of 2 and the number of levels is 1 [419]; a tree is set sequential if and only if it is set graceful [419]; the nontrivial plane triangular grid graph G_{n} is set graceful if and only if n = 2 [421]; every graph can be embedded as an induced subgraph of a connected set sequential graph [419]; every graph can be embedded as an induced subgraph of a connected set graceful graph [419], every planar graph can be embedded as an induced subgraph of a set sequential planar graph [421]; every tree can be embedded as an induced subgraph of a set sequential tree [421]; and every tree can be embedded as an induced subgraph of a set graceful tree [421]. Hegde conjectures [421] that no path is set sequential.

Acharya [13] has shown: C_{n} is set graceful if and only if n has the form 2^{m} − 1 for some m ≥ 2; a connected set graceful graph with q edges and q + 1 vertices is a tree of order p = 2^{m} and for every positive integer m such a tree exists; if G is a connected set sequential graph, then G + K_{1} is set graceful; and if a graph with p vertices and q edges is set sequential, then p + q = 2^{m} − 1. Acharya, Germina, Princy, and Rao [19] proved: if G is set graceful, then G ∪ \overline{K_{t}} is set sequential for some t; if G is a set graceful graph with n edges and n + 1 vertices, then G + \overline{K_{t}} is set graceful if and only if m has the form 2^{t} − 1; P_{n} + \overline{K_{m}} is set graceful if n = 1 or 2 and m has the form 2^{t} − 1; K_{1,m,n} is set graceful if and only if m has the form 2^{t} − 1 and n has the form 2^{s} − 1; P_{4} + \overline{K_{m}} is not set graceful when m has the form 2^{t} − 1 (t ≥ 1); K_{3,5} is not set graceful; if G is set graceful, then graph obtained from G by adding for each vertex v in G a new vertex v′ that is adjacent to every vertex adjacent to v is not set graceful; and K_{3,5} is not set graceful.

7.20 Divisor Graphs

G. Santhosh and G. Singh [749] call a graph G(V, E) a divisor graph if V is a set of integers and uv ∈ E if and only if u divides v or vice versa. They prove the following are divisor graphs: trees; mK_{n}; induced subgraphs of divisor graphs; cocktail party graphs H_{m,n} (see §7.1 for the definition); the one-point union of complete graphs of different orders; complete bipartite graphs; W_{n} for n even and n > 2; and P_{n} + \overline{K}_{t}. They also prove that C_{n} (n ≥ 4) is a divisor graph if and only if n is even and if G is a divisor graph then for all n so is G + K_{n}.

7.21 Difference Graphs

Analogous to a sum graph, Harary [399] calls a graph a difference graph if there is an bijection f from V to a set of positive integers S such that xy ∈ E if and only if |f(x) − f(y)| ∈ S. Bloom, Hell, and Taylor [178] have shown that the following graphs are difference graphs: trees, C_{n}, K_{n}, K_{n,n}, K_{n,n−1}, pyramids, and n-prisms. Gervacio [363] proved that wheels W_{n} are difference graphs if and only if n = 3, 4, or 6. Sonntag [866] proved that cacti (that is, graphs in which every edge is contained in at most one cycle) with girth at least 6 are difference graphs and he conjectures that all cacti are difference graphs.

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Index

(a, r)-geometric, 156 (k, d)-graceful labeling, 39 (m, n)-gon star, 136 A-cordial graph, 53 A-magic, 102 B(n, r, m), 16 B_{n}, 14 C-magic, 104 C_{n}^{(t)}, 11 E_{k}-cordial, 52 G ⊙ H, 13 G_{1}[G_{2}], 15 H-cordial, 52 H-magic, 92 H-supermagic, 92 H_{n}-cordial, 52 K_{n}^{(m)}, 16 M_{n}, 14 P (n, k), 20 P_{n}^{k}, 20 R-ring-magic, 103 T_{p}-tree, 39 W (t, n), 9 α-labeling

eventually, 33 free, 35 near, 35 strong, 34 weakly, 34, 47

α-labeling, 11, 31, 42, 47 α-size, 34 α-valuation, 31 β-valuation, 5 γ-labeling, 46 ρ̂-labelings, 44 ρ-labeling, 44 ρ-valuation, 44 ρ^{+}-labeling, 45 θ-labeling, 45 ρ̃-labelings, 47 a-vertex consecutive magic labeling, 103 b-edge consecutive magic labeling, 104 f-permutation graph, 22 k-cordial labeling, 53 k-fold, 80 k-graceful, 38 kC_{n}-snake, 12, 44

linear, 12 mG, 17 n-cone, 9 n-cube, 15, 31 n-point suspension, 9 t-ply graph, 51 y-tree, 8

mean graph, 158 path-block chain, 22

abbreviated double tree of T, 70 adjacency matrix, 45 almost graceful labeling, 44 almost-bipartite graph, 46 antiprism, 107, 124 apex, 10 arbitrarily graceful, 38 arithmetic, 62

balance index set, 56 bamboo tree, 7, 43 banana tree, 8, 43, 47 bicomposition, 45 bigraceful graph, 22 bipartite labeling, 34 bisequential graph, 61 bistar, 75, 77 block, 79 block-cut-vertex graph, 79 block-cutpoint, 33 block-cutpoint graph, 12 book, 5, 11, 14, 42, 75, 78, 137

generalized, 137

stacked, 15 boundary value, 33

cactus k-angular, 48 triangular, 12

Cartesian product, 13, 48, 49, 143 caterpillar, 7, 31, 35, 42, 47, 59, 76 chain graph, 33, 79 chain of cycles, 11 chord, 10, 137 circulant graph, 68 closed helm, 9 combination graph, 159 complete

n-partite graph, 49, 129 bipartite graph, 12, 15 graph, 15 tripartite graph, 15

component, 138 composition, 15, 48, 143 convex polytope, 99, 123 cordial graph, 49 cordial labeling, 48 corona, 13, 75 critical number, 33 crown, 13, 42, 58, 59, 130, 137, 152 cube, 14, 23 cubic graph, 80 cycle, 5, 44, 133, 137 cycle with a P_{k}-chord, 10 cycle with parallel P_{k} chords, 10 cyclic G-decomposition, 35 cyclic decomposition, 44 cylinders, 99

decomposition, 5, 31, 35, 43, 46 deficiency

edge-magic, 78 super edge-magic, 79

difference graph, 162 disjoint union, 17, 42 divisor graph, 162 dodecahedron, 23 double star, 70 double tree, 70 dragon, 11 Dutch t-windmill, 11 Dutch windmill, 69

edge amalgamation, 137 edge irregular total labeling, 160 edge magic strength, 69 edge reduced

integral sum number, 132 sum number, 132

edge-decomposition, 35 edge-graceful spectrum, 145 elegant labeling, 63 elem. parallel transformation, 39 Eulerian graph, 57 exclusive sum labeling, 134 exclusive sum number, 134

face, 99, 123 fan, 48, 63, 67, 74, 75, 78, 90, 99, 136 firecracker, 8 flag, 50 flower, 9, 136 forest, 79 free α-labeling, 35 friendly index set, 55 friendship graph, 11, 48, 78, 90, 99, 134 full friendly index set, 56 functional extension, 71

gear graph, 9 generalized

book, 137 bundle, 52 fan, 52 wheel, 52

generalized antiprism, 117 generalized caterpillar, 22 generalized helm, 90 generalized Jahangir graph, 90 generalized spider, 22 generalized web, 9, 90 graceful graph, 5 gracesize, 34 gracious k-labeling, 35 gracious labeling, 35 graph

(ω, k)-antimagic, 105 (a, d)-antimagic, 106 (a, r)-geometric, 156 (k, d)-balanced, 40 A-cordial, 53 E-cordial, 148 E_{k}-cordial, 52 H-cordial, 52 H-elegant, 65 H-harmonious, 64 H_{n}-cordial, 52 Δ-optimum summable, 134 θ-Petersen, 146 f-permutation, 22 g-graph, 133 k-antimagic, 105 k-balanced, 58 k-magic, 70 m-level wheel, 146 t-uniform homeomorph, 51 almost-bipartite, 46 antimagic, 105 arbitrarily graceful, 38 arithmetic, 62 bicomposition, 45 bigraceful, 22 bisequential, 61 butterfly, 145 circulant, 68 complete, 15 composition, 15 cordial, 78 countable infinite, 74 diamond, 38 directed, 6 disconnected, 17 divisor, 162 dumbbell, 145 edge-magic, 80 even edge-graceful, 147 even-multiple subdivision, 50 generalized caterpillar, 22 generalized helm, 90 generalized Jahangir graph, 90 generalized spider, 22 generalized web, 9, 90 graceful, 5 graph-block chain, 21 Hamming-graceful, 59 harmonious, 5 hyper strongly multiplicative, 157 ideal magic, 76 join, 19 kite, 11, 77 Knödel, 91 line-graceful, 152 lotus, 38 minimally k-equitable, 58 mirror graph, 15 multiple shell, 57 node-graceful, 41 odd (a, d)-antimagic, 107 odd antimagic, 107 path-block chain, 22 prime, 136 pseudo-magic, 69 pyramid, 38 replicated, 22 semi-magic, 67 set graceful, 161 set sequential, 161 shell-type, 10 simply sequential, 154 Skolem labeled, 42 Skolem-graceful, 41 splitting, 22 strong magic, 76 strongly c-elegant, 66 strongly k-indexable, 78 strongly 1-harmonious, 78 strongly felicitous, 66 strongly multiplicative, 157 supermagic, 67 tadpoles, 11 theta, 64 total, 22, 155 totally magic, 93 unicyclic, 9, 47 uniformly balanced, 54 uniformly cordial, 54 weak magic, 76

graph labeling, 5 graph-block chain, 21 graphs, 159

sunflower, 50 grid, 13, 38

hairy cycle, 13 Hamming-graceful graph, 59 harmonious graph, 5 Heawood graph, 23, 35 helm, 9, 50, 136

closed, 50 generalized, 50

Herschel graph, 23, 106 hexagonal lattice, 99 holey α-labeling, 44 honeycomb graph, 124 host graph, 33 hypercycle, 132

strong, 132 hypergraph, 69, 80, 105, 132 hyperwheel, 132

IC-coloring, 154 IC-index, 154 icosahedron, 23 integer-magic spectrum, 70, 102 integral radius, 132 integral sum

graphs, 130 number, 131 tree, 130

kite, 11, 77 kites, 92

labeling (ω, k)-antimagic, 105 (a, b)-consecutive, 145 (a, d)- vertex-antimagic edge, 106 (a, d)-edge-antimagic total, 112 (a, d)-edge-antimagic vertex, 112 (a, d)-face antimagic, 123 (a, d)-indexable, 112 (a, d)-vertex-antimagic total, 112 (a, r)-geometric, 156 (k, d)-arithmetic, 61 (k, d)-graceful, 39 (k, d)-odd mean, 158 (k, d)-super mean, 158 A-magic, 102 E-cordial, 148 P (a)Q(1)-super vertex-graceful, 148 Q(a)P (b)-super edge-graceful, 148 R-ring-magic, 103 Δ-exclusive sum labeling, 134 Θ-graceful, 48 α-, 31 σ-, 45 a-vertex-consecutive magic, 93 d-antimagic of type (1, 1, 1), 123 k-antimagic, 105 k-cordial, 53 k-edge graceful, 145 k-equitable, 57, 58 k-odd mean, 158 k-sequential, 154 k-sequentially additive, 157 k-super edge mean, 158 k-super mean, 158 1-vertex magic vertex, 103 (k, d)-indexable, 62 k-indexable, 63 strongly (k, d)-indexable, 63 additively (k, d)-sequential, 157 additively graceful, 62 almost graceful, 44 antimagic, 105 balanced, 31, 54 bigraceful, 35 bipartite, 34 complete k-equitable, 59 consecutive, 66 cordial, 48 cordial edge deficiency, 54 cordial vertex deficiency, 54 edge irregular total, 160 edge-graceful, 143 edge-magic, 80 edge-magic total, 74 elegant, 63 felicitous, 65 friendly, 54 gracious, 35 indexable, 63 interlaced, 31 line-graceful, 152 magic, 67, 69, 74

consecutive, 99

of type (0,1,1), 99 of type (1,0,0), 99 of type (1,1,0), 99 of type (1,1,1), 99

mean, 158 near-elegant, 64 nearly graceful, 44 odd sequential, 60 odd-graceful, 42 one modulo three graceful, 47 optimal k-equitable, 59 optimal sum graph, 129 polychrome, 65 prime, 136 prime cordial, 156 product antimagic, 128 product cordial labeling, 155 product edge-antimagic, 128 product edge-magic, 128 product magic, 128 pseudograceful, 46 radio graceful, 152 range-relaxed graceful, 47 semi-elegant, 64 sequential, 59 sigma, 160 simply sequential, 154 Skolem-graceful, 41 strongly c-harmonious, 59 strongly balanced, 54 strongly edge-magic, 76 strongly graceful, 34 strongly harmonious, 61 strongly indexable, 63 strongly super edge-graceful, 148 strongly vertex-magic total, 92 sum graph, 129 super (a, d)-vertex-antimagic total, 112 super edge-antimagic total, 114 super edge-graceful, 146 super edge-magic, 76 super vertex-graceful, 147, 148 super vertex-magic total, 91 supermagic, 67, 78 total magic cordial, 103 total product cordial labeling, 155 totally magic, 93 triangular graceful, 47 vertex irregular total, 160 vertex prime, 138 vertex-magic total, 89 vertex-relaxed graceful, 47

labeling number, 33

lableing

even sequential harmonious, 61 ladder, 13, 47, 60, 99, 137 level joined planar grid, 63 lobster, 7, 43, 44, 47 lotus inside a circle, 99

Möbius ladder, 14, 60, 67, 69, 99, 136,

144 Móbius grid, 118 magic

b-edge consecutive, 93 magic constant, 74 magic square, 67 magic strength, 69, 75 mirror graph, 15 mod integral sum graph, 133 mod integral sum number, 133 mod sum graph, 133 mod sum number, 133 mod sum* graph, 134 mod sum* number, 134 Mongolian tent, 13, 38 Mongolian village, 14, 39 multigraph, 78, 80 multiple shell, 10 mutation, 92

near α-labeling, 35 nearly graceful labeling, 44

odd-graceful labeling, 42 olive tree, 7 one modulo three graceful labeling, 47 one-point union, 11, 16, 31, 42, 48, 49,

65, 137 optimal sum graph, 129

parachutes, 106 parallel chord, 55 path, 10, 63 path-union, 51 pendant edge, 34 perfect system of difference sets, 39 permutation graph, 159 Petersen graph, 23

generalized, 20, 49, 75, 78, 89, 106,

112, 113 planar bipyramid, 99 planar graph, 99, 123 Platonic family, 99 polyminoes, 38 prime cordial labeling, 156 prime graph, 136 prime labeling, 136 prism, 14, 89, 99, 112, 123 product cordial, 155 product cordial labeling, 155 product graph, 129 pseudo-magic graph, 69 pseudograceful labeling, 46

radio graceful, 152 radio labeling, 151 radio number, 151 range-relaxed graceful labeling, 47 real sum graph, 129 regular graph, 67, 68, 75, 90, 103 replicated graph, 22 representation, 153 representation number, 153 root, 50 root-union, 56 rooted tree, 42

saturated vertex, 130 sequential join, 33 shell, 10, 50

multiple, 10 shell graph, 53 sigma labeling, 160 Skolem labeled graph, 42 Skolem sequence, 7, 18 Skolem-graceful labelings, 41 snake, 12

n-polygonal, 12, 48 double triangular, 12 quadrilateral, 32 triangular, 44 spanning tree, 46 sparse semi-magic square, 93 spider, 7 splitting graph, 22, 43 spum, 129 stable set, 22 star, 19, 20, 41, 47, 90, 152 strength

edge magic, 69 magic, 69, 75

maximum magic, 76 strong

sum graph, 130 strong edge graceful

l, 145 strong vertex-graceful, 147 strongly ⋆-graph, 159 strongly graceful labeling, 34 stunted tree, 45 subdivision, 7, 13, 43, 99 sum graph, 129

mod, 133

mod integral, 133 real, 129

sum number, 129 sum* graph, 134 sum* number, 134 sunflower, 50, 144 super magic strength, 77 supersubdivision, 21 symmetric product, 15, 44

tadpoles, 11 theta graph, 64 torus grid, 14 total edge (vertex) irregular strength, 160 total edge irregularity strength, 160 total graph, 22, 56, 155 total product cordial, 155 total product cordial labeling, 155 tree, 5, 7, 74, 78, 80, 90, 105, 133

binary, 75 symmetrical, 7

triangular graceful labeling, 47 twig, 137

unicyclic graph, 11, 47 union, 17, 18, 35, 74, 77, 78, 90, 130, 136,

138

vertex irregular total labeling, 160 vertex prime labeling, 138 vertex-graceful, 147 vertex-relaxed graceful labeling, 47 weak tensor product, 33, 35 weakly α-labeling, 34 web, 9

generalized, 76 weight, 123 wheel, 9, 48, 50, 59, 67, 74, 90, 99, 105,

113 windmill, 16, 50 working vertex, 134 wreath product, 65

Young tableau, 14, 39