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 ? ' a ( G ) . In this paper, we prove that ? ' a ( G ) ≤ 5 2 ( 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 ? ' a ( G ) ≤ 3 Δ for any graph G without isolated edges.

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