On acyclic 4-choosability of planar graphs without short cycles

Let G = (V, E) be a graph. A proper vertex coloring of G is acyclic if G contains no bicolored cycle. Namely, every cycle of G must be colored with at least three colors. G is acyclically L-list colorable if for a given list assignment L = {L(v) : v is an element of V}, there exists a proper acyclic coloring pi of G such that pi (v) is an element of L(v) for all v is an element of V. If G is acyclically L-list colorable for any list assignment with |L(v)| >= k for all v is an element of V. then G is acyclically k-choosable. In this paper, we prove that planar graphs with neither {4, 5}-cycles nor 8-cycles having a triangular chord are acyclically 4-choosable. This implies that planar graphs either without {4, 5, 7}-cycles or without {4, 5, 8}-cycles are acyclically 4-choosable. (C) 2010 Elsevier B.V. All rights reserved.