Some Results Of The Adjacent Vertex Distinguishing Proper Edge Coloring Of Graphs

A proper edge coloring of G is called adjacent vertex distinguishing if any two adjacent vertices u and v are incident to different sets of colored edges. Obviously, a graph G has adjacent vertex distinguishing proper edge coloring if and only if G has no isolated edges. The minimum number required for an adjacent vertex distinguishing proper edge coloring of G is called the adjacent vertex distinguishing proper edge chromatic number, denoted by x'_a(G)- The adjacent vertex distinguishing proper edge chromatic numbers of P_m×P_n, P_m×C_n, P_n~k, monocycle graph and several complete 4—partite graphs are discussed and the adjacent vertex distinguishing proper edge chromatic numbers of them are obtained in this paper. These results illustrate that the adjacent vertex distinguishing proper edge coloring conjecture (For any connected graph G, |V(G)|≥6, we have△(G)≤X'_a(G)≤△(G) + 2) is true for these graphs. And an upper bound of the adjacent vertex distinguishing proper edge chromatic number is given for a class graph withδ≥5 and△<(2(cn+1))/7 , where 0 < c <(7/8).