On acyclic 4-choosability of planar graphs without cycles of length 4, 7 and 9

Let G = (V, E). A proper vertex coloring of G is acyclic if there is no bicolored cycle in G. Given a list assignment L = {L(v) vertical bar v is an element of V} of G, if there exists an acyclic coloring pi of G such that pi(v) is an element of L(v) for all v is an element of V, then we say that G is acyclically L-colorable. If G is acyclically L-colorable for any list assignment L with vertical bar L(v)vertical bar >= k for all v is an element of V, then G is acyclically k-choosable. It is known that every planar graph without {4, i, j}-cycles is acyclically 4-choosable for each pair {i, j} subset of {5, 6, 7, 8, 9} and {i, j} is not an element of{{7, 9}, {8, 9}}. In this paper, we prove that every planar graph with neither 4-cycles, 7-cycles nor chordal 9-cycles is acyclically 4-choosable. As a corollary, every planar graph without {4, 7, 9}-cycles is acyclically 4-choosable. (C) 2021 Elsevier B.V. All rights reserved.