Adjacent vertex-distinguishing edge coloring of graphs with maximum degree at least five

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let mad(G) and Δ(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this note, we prove that every graph G with Δ(G)⩾5 and mad(G)<135 can be avd-colored with Δ(G)+1 colors. This strengthens a result of Wang and Wang [W. Wang and Y. Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree, J. Comb. Optim., 19:471-485, 2010].