Acyclic 6-choosability of Planar Graphs without 5-cycles and Adjacent 4-cycles

A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors. A graph G is acyclically k-choosable if for any list assignment L = { L ( v ): v ∈ V ( G )} with ∣ L ( v )∣ ≥ k for all v ∈ V ( G ), there exists a proper acyclic vertex coloring φ of G such that φ( v ) ∈ L ( v ) for all v ∈ V ( G ). In this paper, we prove that if G is a planar graph and contains no 5-cycles and no adjacent 4-cycles, then G is acyclically 6-choosable.