Vertex coloring complete multipartite graphs from random lists of size 2

Let K-sxm be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K-sxm a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size sigma(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring phi, such that phi(v) is an element of L(v) for all v is an element of V(K-sxm). We show that this property exhibits a sharp threshold at sigma(m) = 2(s - 1)m. (C) 2010 Elsevier B.V. All rights reserved.