Adapted list coloring of planar graphs

Abstract

AbstractGiven an edge coloring F of a graph G, a vertex coloring of G is adapted to F if no color appears at the same time on an edge and on its two endpoints. If for some integer k, a graph G is such that given any list assignment L to the vertices of G, with |L(v)|⩾k for all v, and any edge coloring F of G, G admits a coloring c adapted to F where c(v)∈L(v) for all v, then G is said to be adaptably k‐choosable. In this note, we prove that K5‐minor‐free graphs are adaptably 4‐choosable, which implies that planar graphs are adaptably 4‐colorable and answers a question of Hell and Zhu. We also prove that triangle‐free planar graphs are adaptably 3‐choosable and give negative results on planar graphs without 4‐cycle, planar graphs without 5‐cycle, and planar graphs without triangles at distance t, for any t⩾0. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 127–138, 2009