an:01436155
Zbl 0944.05040
Cropper, M. M.; Goldwasser, J. L.; Hilton, A. J. W.; Hoffman, D. G.; Johnson, P. D. jun.
Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs
EN
J. Graph Theory 33, No. 4, 199-219 (2000).
00063923
2000
j
05C15
list-colorings; list-multicolorings; clique
Philip Hall's theorem on systems of distinct representatives [J. Lond. Math. Soc. 10, 26-30 (1935; Zbl 0010.34503)] and an improvement by \textit{P. R. Halmos} and \textit{H. E. Vaughan} [Am. J. Math. 72, 214-215 (1950; Zbl 0034.29601)] can be interpreted as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The Hall-Halmos-Vaughan theorem can be stated: if \(G\) is a clique, then Hall's condition is sufficient for the existence of a proper multicoloring. The present authors study the class HHV of simple graphs \(G\) for which Hall's condition guarantees the existence of a proper multicoloring. They also show that HHV is contained in the class of graphs for which every block is a clique and each cut-vertex is in exactly two blocks. For paths, the authors address the problem of deciding if this is a proper coloring and, if so, of finding one.
Arthur T.White (Kalamazoo)
Zbl 0010.34503; Zbl 0034.29601