Vertex coloring complete multipartite graphs from random lists of size 2

Let K s × m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of K s × m a list L ( v ) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ ( m ) . In this paper we determine, for all fixed s and growing m , the asymptotic probability of the existence of a proper coloring φ , such that φ ( v ) ∈ L ( v ) for all v ∈ V ( K s × m ) . We show that this property exhibits a sharp threshold at σ ( m ) = 2 ( s − 1 ) m .