D(β)-Vertex-distinguishing Edge-Coloring Of Graphs And The Probabilistic Method

In 2006,Zhang Zhongfu present a new concept of the D(β)-vertex-distinguishing proper edge-coloring and conjecture,see[8].In this pa-per,we has been studied D(β)-vertex-distinguishing proper edge-coloring by probabilistic and giving colorings of a graph.The paper consists of five part.In the first part,we introduce some fundamental concepts that often used in the paper.In the second part,we discuss the properties of D(β)-vertex-distinguishing proper edge-coloring and,we has been obtained X'_3-vd(P_n) and X'_3-vd(C_n),by induction we has gotten the D(β)-vertex-distinguishing proper edge-coloring chromatic number of P_n,S_n,C_n,K_n,F_n,W_nIn the third part,we have discussed and obtained the D(β)-vertex-distinguishing proper edge-coloring chromatic number of S_m∨P_n∨P_n,F_m∨P_n with giving coloring of a graph.In the fourth part,we have discussed and obtained the D(β)-vertex-distinguishing proper edge-coloring chromatic number of C_m·P_n,C_m·S_n,C_m·C_n,C_m·F_n,C_m·W_n with giving coloring of a graph,whenβ=1,2,3.In the last part,With applying the general form of the Lovász lo-cal lemma,we have discussed and obtained that X'_2-vd(G)≤32d²,d≥4,X'_3-vd(G)≤8d^5/2,d≥6 and whenβ≥4,d≥4,X'_β-vd(G)≤2(?)for any simple graph G with maximum degree d.we has gotten another upper bounds for D(2)-vertex-distinguishing proper edge-coloring chromatic number X'_2-vd(G)≤△+1+(?)for the simple graph with maximum degree△≥16² ln 6,minimum degreeδ≥16(?) With applying the general form of the Lovász local lemma by another method...