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Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs. (English) Zbl 0944.05040
Philip 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
list-colorings; list-multicolorings; clique
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##### References:
 [1] Hall, J London Math Soc 10 pp 26– (1935) · Zbl 0010.34503 · doi:10.1112/jlms/s1-10.37.26 [2] Halmos, Am J Math 72 pp 214– (1950) · Zbl 0034.29601 · doi:10.2307/2372148 [3] and ?A variation of Ryser’s theorem and a necessary condition for the list-colouring problem,? Graph colourings, and (Editors), Pitman Research Notes in Mathematics Series 218, Longman Scientific and Technical, Wiley, Harlow, Essex, and New York, 1990, pp. 135-143. [4] and Extending Hall’s theorem, Topics in combinatorics and graph theory; Essays in honour of Gerhard Ringel, Physica-Verlag, Heidelberg, pp. 360-371. [5] Hilton, Congress Numer 121 pp 161– (1996) [6] Hilton, Congress Numer 128 pp 195– (1997) [7] Introduction to graph theory, Prentice Hall Upper Saddle River, NJ, 1996. · Zbl 0891.05001
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