One upper bound on the D(3)-vertex-distinguishing edge- chromatic numbers of graphs

Let G(V,E) be a connected graph with order ≥ 3, α β be positive integers, suppose mapping from E(G) to {1, 2, ..., α} is a proper edge coloring ∀x ∈ V (G). Let S(x) be the set of colors of the all edges incident to x, S(u) ≠ S(v) whenever u, v ∈ V (G) with 1 ≤ d(u, v) ≤ β, then f is called an α-D(β)-vertex-distinguishing proper edge-coloring of graph G(Shortly, an α-D(β)-VDPEC of G) and the number χ_β-vd´(G) = min{α | G has an α-D(β)-VDPEC} is called the D(β)-vertex-distinguishing edge-chromatic number of G. In this paper, let d be the maximum degree of G, we study the upper bound for the D(3)-vertex-distinguishing edge-chromatic number by probability method and prove that χ_3-vd´(G)≤ 4(2d⁵-d⁴+8d³-9d²+6d-1/d-1), d ≥ 2.