### On the acyclic choosability of graphs

Mickaël Montassier, Pascal Ochem, André Raspaud
2006 Journal of Graph Theory
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 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 M ad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic
more » ... loring, choosability and acyclic choosability. Borodin, Kostochka and Woodall improved this bound for planar graphs with a given girth. We recall that the girth of a graph is the length of its shortest cycle. Theorem 2 [BKW99] 1. Every planar graph with girth at least 7 is acyclically 3-colorable. Every planar graph with girth at least 5 is acyclically 4-colorable. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V (G)} there exists a coloring c of the vertices such that c(v) ∈ L(v) and c(v) = c(u) if u and v are adjacent in G. If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V (G), then G is said k-choosable. In [Tho94], Thomassen proved that every planar graph is 5-choosable and Voigt proved that there are planar graphs which are not 4-choosable [Voi93]. In the following, we are interested in the acyclic choosability of graphs. In [BFDFK + 02], the following theorem is proved and the next conjecture is given : Theorem 3 [BFDFK + 02] Every planar graph is acyclically 7-choosable. This means that for any given list assignment L such that ∀v ∈ V, |L(v)| ≥ 7, we can choose for each vertex v a color in L(v) such that the obtained coloring of G is acyclic. The acyclic list chromatic number of G, χ l a (G), is the smallest integer k such that G is acyclically k-choosable. Conjecture 1 [BFDFK + 02] Every planar graph is acyclically 5-choosable. Conjecture 1 is very strong, since it implies the celebrated result of Borodin (Theorem 1), and we know that its proof is tough. A first observation can be made concerning outerplanar graphs : Proposition 1 Every outerplanar graph is acyclically 3-choosable.