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) ≤ ∆ + 2, where ∆ = ∆(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 ∆ + 2 colors to be acyclically edge colored. In this paper, using a simple counting argument we observe not only that this is not true, but infact all d-regular graphs with 2n vertices and d > n, requires at least d + 2 colors. We also show that a(Kn,n) ≥ n + 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 d ≥ 5, n ≥ 2d + 3 and dn even, there exist d-regular graphs which require at least d+ 2-colors to be acyclically edge colored.