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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 $χ_a'(G)$ <= 5($Δ+2$)/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 $χ_a'(G)$ <= 3$Δ$ for any graph G without isolated edges.