International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 72168, 17 pages
doi:10.1155/2007/72168
Extending Hall's theorem into list colorings: a partial history
D.G. Hoffman
and P.D.jun. Johnson
Department of Mathematics and Statistics, Auburn University, Auburn 36849, AL, USA
Abstract
In 1988, A. J. W. Hilton and P. D. Johnson Jr. found a natural generalization of the condition in Philip Hall's celebrated theorem on systems of distinct representatives. This generalization was formed in the relatively new theory of list colorings of graphs. Here we give an account of a strand of development arising from this generalization, concentrating on extensions of Hall's theorem. New results are presented concerning list colorings of independence systems and colorings of graphs with nonnegative measurable functions on positive measure spaces.