Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs

Abstract

Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall-Halmos-Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut-vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut-vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000