Vertex coloring complete multipartite graphs from random lists of size 2

Abstract

Let $K_{s\times m}$ be the complete multipartite graph with $s$ parts and $m$ vertices in each part. Assign to each vertex $v$ of $K_{s\times m}$ a list $L\left(v\right)$ of colors, by choosing each list uniformly at random from all 2-subsets of a color set $\mathcal{C}$ of size $\sigma \left(m\right)$. 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 $\phi \left(v\right)\in L\left(v\right)$ for all $v\in V\left(K_{s\times m}\right)$. We show that this property exhibits a sharp threshold at $\sigma \left(m\right)=2(s-1)m$.