Random Graph-Homomorphisms and Logarithmic Degree
Itai Benjamini, Weizmann Institute
Ariel Yadin, Weizmann Institute
Amir Yehudayoff, Weizmann Institute
Abstract
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant.
Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs Cn,k (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function ψ(n) tending to infinity, the range of a typical homomorphism of Cn,k is super-constant for k = 2 log(n) - ψ(n), and is 3 for k = 2 log(n) + ψ(n)
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