Acyclic 5-choosability of planar graphs without 4-cycles

Abstract

A proper vertex coloring of a graph G = ( V , E ) is acyclic if G contains no bicolored cycle. A graph G is acyclically L -list colorable if for a given list assignment L = { L ( v ) : v ∈ V } , there exists a proper acyclic coloring π of G such that π ( v ) ∈ L ( v ) for all v ∈ V . If G is acyclically L -list colorable for any list assignment with | L ( v ) | ≥ k for all v ∈ V , then G is acyclically k -choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281–300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.