Acyclic 4-choosability of planar graphs without intersecting short cycles

Abstract

A proper vertex coloring of [Formula: see text] is acyclic if [Formula: see text] contains no bicolored cycle. Namely, every cycle of [Formula: see text] must be colored with at least three colors. [Formula: see text] is acyclically [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an acyclic coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is acyclically [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is acyclically [Formula: see text]-choosable. In this paper, we prove that planar graphs without intersecting [Formula: see text]-cycles are acyclically [Formula: see text]-choosable. This provides a sufficient condition for planar graphs to be acyclically 4-choosable and also strengthens a result in [M. Montassier, A. Raspaud and W. Wang, Acyclic 4-choosability of planar graphs without cycles of specific lengths, in Topics in Discrete Mathematics, Algorithms and Combinatorics, Vol. 26 (Springer, Berlin, 2006), pp. 473–491] which says that planar graphs without [Formula: see text]-, [Formula: see text]-cycles and intersecting 3-cycles are acyclically 4-choosable.