On strong edge-coloring of graphs with maximum degree 4

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 that for any simple and finite graph G, a(G) ≤ ∆+ 2, where ∆ = ∆(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with ∆(G) ≤ 4, with the additional restriction that m ≤ 2n− 1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m ≤ 2n, when ∆(G) ≤ 4. It follows that for any graph G if ∆(G) ≤ 4, then a(G) ≤ 7.