Acyclic 4-choosability of planar graphs

A proper vertex coloring of a graph G = (V , E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v) | v ∈ V } of G, we say G is acyclically L-list colorable if 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 paperwe prove that planar graphswithout 4, 7, and 8-cycles are acyclically 4-choosable. © 2010 Elsevier B.V. All rights reserved.