Equitable list coloring of graphs

A graph G is equitably k-choosable if, for any k-uniform list assignment L, G admits a proper coloring pi such that pi(nu) is an element of L(nu) for all nu is an element of V(G) and each color appears on at most [\G\/k] vertices. It was conjectured in [8] that every graph G with maximum degree Delta is equitably k-choosable whenever k greater than or equal to Delta + 1. We prove the conjecture for the following cases: (i) Delta less than or equal to 3; (ii) k greater than or equal to (Delta - 1)(2). Moreover, equitably 2-choosable graphs are completely characterized.