Acyclic choosability of graphs with small maximum degree Daniel Gonçalves ⋆ and Mickaël Montassier ⋆⋆ LaBRI UMR CNRS 5800, Université Bordeaux I, 33405 Talence Cedex FRANCE. Abstract A proper vertex coloring of a graph G =(V,E) is acyclic if G con- tains no bicolored cycle. A graph G is L-list colorable if for a given list assign- ment 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 coloring obtained is acyclic. In this paper, we study the acyclic choosability of graphs with small maximum degree. In 1979, Burstein proved that every graph with maximum degree 4 admits a proper acyclic color- ing using 5 colors [Bur79]. We prove that (a) every graph with maximum degree Δ =3 is acyclically 4-choosable and (b) every graph with maximum degree Δ =4 is acyclically 5-choosable. The proof of (b) uses a backtracking greedy algorithm and Burstein’s theorem. 1 Introduction Let G be a graph. Let V (G) be its set of vertices and E(G) be its set of edges. A proper vertex coloring of G is an assignment f of integers (or labels) to the vertices of G such that f (u) ̸= f (v) if the vertices u and v are adjacent in G.A k-coloring is a proper vertex coloring using k colors. A proper vertex coloring of a graph is acyclic if there is no bicolored cycle. The acyclic chromatic number of G, χ a (G), is the smallest integer k such that G is acyclically k-colorable. Acyclic colorings were introduced by Grünbaum in [Grü73] and studied by Mitchem [Mit74], Albertson, Berman [AB77], and Kostochka [Kos76]. In 1979, Borodin proved Grünbaum’s conjecture: Theorem 1. [Bor79] Every planar graph is acyclically 5-colorable. This bound is best possible: in 1973, Grünbaum gave an example of a 4-regular planar graph [Grü73] which is not acyclically colorable with four colors. Moreover, there exist bipartite 2-degenerate planar graphs which are not acyclically 4-colorable [KM76]. ⋆ goncalve@labri.fr ⋆⋆ montassi@labri.fr