On the D(β)-vertex-distinguishing acyclic edge chromatic number of graphs

A proper k-edge coloring of a graph G is called D(β)-vertex-distinguishing acyclic edge coloring, if (1) there is no 2-colored cycle in G, and (2) any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the edges incident this vertex. In this paper, we study D(2)-vertex-distinguishing acyclic edge coloring of some special graphs-the cycle C_n, the fan F_n, the wheel W_n, the complete bipartite graph K_{m, n}, and the complete graph K_n. And prove that if G(V, E) is a graph with Δ(G) ≥ 4, and without isolated edges, then X_2-vda(G) ≤ 8Δ².