EQUITABLE LIST COLORING OF GRAPHS

Abstract

A graph $G$ is equitably $k$-choosable if, for any $k$-uniform list assignment $L$, $G$ admits a proper coloring $\pi$ such that $\pi(v)\in L(v)$ for all $v\in V(G)$ and each color appears on at most $\lceil |G|/k\rceil$ vertices. It was conjectured in [8] that every graph $G$ with maximum degree $\Delta$ is equitably $k$-choosable whenever $k\ge \Delta+1$. We prove the conjecture for the following cases: (i) $\Delta \le 3$; (ii) $k\ge (\Delta-1)^2$. Moreover, equitably 2-choosable graphs are completely characterized.