Acyclic choosability of planar graphs: a Steinberg like approach

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∈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 ϕ of G such that ϕ(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment L with |L(v)|⩾k for all v∈V(G), then G is acyclically k-choosable. In this note, we prove that every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable.