Acyclic 4-choosability of Planar Graphs with Girth at Least 5

Abstract

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable, for a given list assignment L = L(v): v ∈ V, if there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ κ for all v ∈ V, then G is called κ-choosable. A graph is said to be acyclically κ-choosable if these L-list colorings can be chosen to be acyclic. In this paper, we prove that if G is planar with girth g ≥ 5, then G is acyclically 4-choosable. This improves the result of Borodin, Kostochka and Woodall [[BKW99]] concerning the acyclic chromatic number of planar graphs with girth at least 5.