Choice Numbers of Graphs: a Probabilistic Approach

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 ≥ 2 and r ≥ 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 Erdős, 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 ( n 1/2 (log n ) 1/2 ).