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 paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.