On acyclic 4-choosability of planar graphs without short cycles

AbstractLet 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 acyclicallyL-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 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