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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
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References:
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