An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by χ′a(G). In this paper, we prove that χ′a(G)⩽52(Δ+2) for any graph G having maximum degree Δ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-strong edge colorings of graphs, Discrete Math. 306 (2006) 3005-3010], which states that χ′a(G)⩽3Δ for any graph G without isolated edges.