Every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable
An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v is an element of V(G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring phi of G such that phi(v) is an element of L(v) for all v is an element of V(G). If G is acyclically L-list colorable for any list assignment L with vertical bar L(v)vertical bar >= k for all v is an element of V(G), then G is acyclically k-choosable. In this paper, we prove that every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable. (C) 2009 Elsevier B.V. All rights reserved.
planar graphs combinatorial problems acyclic choosability forbidden cycles