d-Regular graphs of acyclic chromatic index at least d+2

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a′(G)lΔ+2, where Δ=Δ(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Δ+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d>n, requires at least d+ 2 colors. We also show that a′(Kn, n)gn+ 2, when n is odd using a more non-trivial argument. (Here Kn, n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn, n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that dg5, ng2d+ 3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226–230, 2010