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.