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International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 72168, 17 pages
http://dx.doi.org/10.1155/2007/72168
Review Article

Extending Hall's Theorem into List Colorings: A Partial History

Department of Mathematics and Statistics, Auburn University, Auburn 36849, AL, USA

Received 14 September 2006; Accepted 12 February 2007

Academic Editor: Eugene H. Dshalalow

Copyright © 2007 D. G. Hoffman and P. D. Johnson Jr. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.