an:00563588
Zbl 0793.05076
Alon, Noga
Choice numbers of graphs: A probabilistic approach
EN
Comb. Probab. Comput. 1, No. 2, 107-114 (1992).
00018570
1992
j
05C35 05C15
choice number; coloring
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 \textit{P. Erd??s}, \textit{A. L. Rubin} and \textit{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 \(O(n^{1/2}(\log n)^{1/2})\).
Zbl 0469.05032