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

*(English)*Zbl 0944.05040Philip Hall’s theorem on systems of distinct representatives [J. Lond. Math. Soc. 10, 26-30 (1935; Zbl 0010.34503)] and an improvement by P. R. Halmos and 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.

Reviewer: Arthur T.White (Kalamazoo)

##### MSC:

05C15 | Coloring of graphs and hypergraphs |

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\textit{M. M. Cropper} et al., J. Graph Theory 33, No. 4, 199--219 (2000; Zbl 0944.05040)

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##### References:

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[2] | Halmos, Am J Math 72 pp 214– (1950) · Zbl 0034.29601 · doi:10.2307/2372148 |

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