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Choice numbers of graphs: A probabilistic approach. (English) Zbl 0793.05076
Summary: The choice number of a graph $$G$$ is the minimum integer $$k$$ such that for every assignment of a set $$S(v)$$ of $$k$$ colors to every vertex $$v$$ of $$G$$, there is a proper coloring of $$G$$ that assigns to each vertex $$v$$ a color from $$S(v)$$. By applying probabilistic methods, it is shown that there are two positive constants $$c_ 1$$ and $$c_ 2$$ such that for all $$m\geq 2$$ and $$r\geq 2$$ the choice number of the complete $$r$$-partite graph with $$m$$ vertices in each vertex class is between $$c_ 1 r\log m$$ and $$c_ 2 r\log m$$. This supplies the solutions of two problems of P. Erdős, A. L. Rubin and H. Taylor [Combinatorics, graph theory and computing, Proc. West Coast Conf., Arcata/Calif. 1979, 125-157 (1980; Zbl 0469.05032)], as it implies that the choice number of almost all the graphs on $$n$$ vertices is $$o(n)$$ and that there is an $$n$$ vertex graph $$G$$ such that the sum of the choice number of $$G$$ with that of its complement is at most $$O(n^{1/2}(\log n)^{1/2})$$.

##### MSC:
 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs
##### Keywords:
choice number; coloring
Full Text:
##### References:
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