An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χ a (G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χ a (G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χ a (G) ≤ Δ(G) + 2 for bipartite graphs. These bounds are tight. For k-chromatic graphs G without isolated edges we prove a weaker result of the form χ a (G) = Δ(G) + O(log k).
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