English

Abstract

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycle in G. In other words, each cycle of G must be colored with at least three colors. Given a list assignment L = {L(v): v ∈ V}, if there exists an acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V, then we say that G is acyclically L-colorable. If G is acyclically L-colorable for any list assignment L with ∣L(v)∣ ⩾ k for all v ∈ V, then G is acyclically k-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting i-cycles for each i ∈ {3, 5} is acyclically 4-choosable.