On list r-hued coloring of planar graphs

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