A sufficient condition for planar graphs to be acyclically 5‐choosable

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 article we prove that every planar graph without 4‐cycles and without intersecting triangles is acyclically 5‐choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4‐cycles and without two triangles at distance less than 3 is acyclically 5‐choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory