On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G is said k-choosable. A graph is said to be acyclically k-choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k-choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability and acyclic choosability.