On list r-hued coloring of planar graphs

Abstract

A list assignment of G is a function L that assigns to each vertex $v\in V(G)$ 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 $\phi$ such that for any vertex v with degree d(v), $\phi (v)\in L(v)$ and v is adjacent to at least $min\{d(v),r\}$ different colors. The list r-hued chromatic number of G, $\chi _{L,r}(G)$, is the least integer k such that for every list assignment L with $|L(v)|=k$, $v\in V(G)$, G has an (L, r)-coloring. We show that if $r\ge 32$ and G is a planar graph without 4-cycles, then $\chi _{L,r}(G)\le r+8$. This result implies that for a planar graph with maximum degree $\varDelta \ge 26$ and without 4-cycles, Wagner’s conjecture in [Graphs with given diameter and coloring problem, Technical Report, University of Dortmund, Germany, 1977] holds.