D(2)-Vertex Distinguishing Proper Edge Coloring And Vertex Distinguishing Proper Edge Coloring Of Graphs

Let G(V,E)be a connect graph with order at least 3,αandβare positive integers, and f is a mapping from E(G)to {1,2,…,α}.If f is a proper edge coloring,and for any positiveβand any u,v∈V(G),0＜d(u,v)≤β,we have C(u)≠C(v),then f is calledα-D(β)-vertex distinguishing proper edge coloring of graph G(α-D(β)-VDPEC of G in brief)and the numberχ'_β-vd(G)=min{α|G has aα-D(β)-VDPEC} is calledα-D(β)-vertex distinguishing edge chromatic number of graph G.It is obvious thatχ'_β-vd(G)exists for any graph G with |V(G)]≥3.Andχ'_1-vd(G)=χ'_as(G)is called adjacent strong edge chromatic number of graph G;χ'_D-vd(G)=χ'_s(G) is called vertex distinguishing edge chromatic number of graph G,where D denotes the diameter of connected graph G.In this paper some problems of D(2)-vertex distinguishing proper edge coloring and vertex distinguishing proper edge coloring are discussed.There are four parts:In the first part,some definitions,theorems and so on are giving.In the second part,D(2)-vertex distinguishing proper edge coloring of extended Mycielski graphs of some graphs are discussed,and the D(2)-vertex distinguishing proper edge chromatic numbers are obtained.In the third part,with applying Lovász Local Lemma-General Local Lemma,an upper bounds for D(2)-vertex distinguishing proper edge chromatic number is discussed and obtained.At last,the vertex-distinguishing proper edge chromatic number of K_n^c∨K_t,P_m∨K_n(m≥5)and C_m∨K_n(m≥4)are also discussed.