Coloring complete bipartite graphs from random lists

Let K-n,K-n be the complete bipartite graph with n vertices in each side. For each vertex draw uniformly at random a list of size k from a base set S of size s = s(n). In this paper we estimate the asymptotic probability of the existence of a proper coloring from the random lists for all fixed values of k and growing n. We show that this property exhibits a sharp threshold for k >= 2 and the location of the threshold is precisely s(n) = 2n for k = 2 and approximately s(n) = 2(k-1) In 2/n for k >= 3. (c) 2005 Wiley Periodicals, Inc.