 
Summary: Choice numbers of graphs; a probabilistic approach
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
and Bellcore, Morristown, NJ 07960, USA
Abstract
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). Applying probabilistic methods it is shown that there are two
positive constants c1 and c2 such that for all m 2 and r 2 the choice number of the complete
rpartite graph with m vertices in each vertex class is between c1r log m and c2r log m. This
supplies the solutions of two problems of Erdos, Rubin and Taylor, 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(n1/2
(log n)1/2
).
Research supported in part by a United States Israel BSF Grant
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