Acyclic 4-Choosability of Planar Graphs Without Cycles of Specific Lengths

Abstract

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 c of G such that c(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.

Let G be a planar graph without 4-cycles and 5-cycles. In this paper, we prove that G is acyclically 4-choosable if G furthermore satisfies one of the following conditions: (1) without 6-cycles; (2) without 7-cycles; (3) without intersecting triangles.

On leave of absence from the Departement of Mathematics, Zhejiang Normal University, Jinhua 321004, P. R. CHINA. Supported by the french CNRS.