On list r-hued coloring of planar graphs

A list assignment of G is a function L that assigns to each vertex a list L(v) of available colors. Let r be a positive integer. For a given list assignment L of G, an (L, r)-coloring of G is a proper coloring such that for any vertex v with degree d(v), and v is adjacent to at least different colors. The list r-hued chromatic number of G, , is the least integer k such that for every list assignment L with , , G has an (L, r)-coloring. We show that if and G is a planar graph without 4-cycles, then . This result implies that for a planar graph with maximum degree and without 4-cycles, Wagner's conjecture in [Graphs with given diameter and coloring problem, Technical Report, University of Dortmund, Germany, 1977] holds.