Acyclic 5-choosability of planar graphs with neither 4-cycles nor chordal 6-cycles

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 epsilon V}, there exists a proper acyclic coloring phi of G Such that phi(v) epsilon L(v) for all v epsilon V(G). If G is acyclically L-list colorable for any list assignment with |L(v)| >= k for all v epsilon V, then G is acyclically k-choosable. In this paper it is proved that every planar graph with neither 4-cycles nor chordal 6-cycles is acyclically 5-choosable. This generalizes the results of [M. Montassier. A. Raspaud, W. Wang, Acyclic 5-choosability of planar graphs without small cycles, J. Graph Theory 54 (2007) 245-260]. and a corollary of [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs,J. Graph Theory 51 (4) (2006) 281-300]. (C) 2009 Elsevier B.V. All rights reserved.