A SUFFICIENT CONDITION FOR ACYCLIC 5-CHOOSABILITY OF PLANAR GRAPHS WITHOUT 5-CYCLES

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L = {L (nu) : nu is an element of V ( G)}, there exists an acyclic coloring phi of G such that phi (nu) is an element of L (nu) for all nu is an element of V (G). A graph G is acyclically k-choosable if G is acyclically L-list colorable for any list assignment with L (nu) >= k for all nu is an element of V (G). Let G be a planar graph without 5-cycles and adjacent 4-cycles. In this article, we prove that G is acyclically 5-choosable if every vertex nu in G is incident with at most one i-cycle, i is an element of {6, 7}.