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.