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 ∈ V } , there exists a proper acyclic coloring ϕ of G such that ϕ ( v ) ∈ L ( v ) for all v ∈ V ( G ) . 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 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].