Abstract: The weak adjacent vertex distinguishing edge coloring of graph G is a proper edge coloring of G such that any pair of adjacent maximum degree vertices have distinct sets of colors. The minimum number of colors required for a weak adjacent vertex distinguishing edge coloring of Gis denoted by \chi _a\Delta ^'(G) . In this paper, we prove that if G is a planar graph without 3-cycles, then \chi _a\Delta ^'(G)\le \mathrmmax\9,\Delta (G)+1\ .