On list r-hued coloring of planar graphs

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.