The Vertex-distinguishing Edge-Coloring And The Vertex-distinguishing Total Coloring Of Graph

The coloring problems is one of popular problems in graph theory because of it's profound significance in both theory and practice. There have been a lot of conclusions about it from the domestic and foreign researchers, but the research involved in "the vertex-distinguishing edge-coloring" or "the vertex-distinguishing total coloring" is actually few because they are so difficult. Zhang Zhongfu defined the D(β)- vertex-distinguishing edge-coloring in 2006. The adjacent strong edge coloring can be looked as D(1)- vertex-distinguishing edge-coloring while the vertex-distinguishing edge-coloring can be looked as D(β)- vertex-distinguishing edge-coloring whenβis the diameter of the graph. The research involved in the adjacent strong edge coloring is more. In view of this background, this paper has done the following works toward some special graphs which have their adjacent strong edge chromatic numbers.Firstly, this paper presents a general overview of the basic concepts and the research status quo of the edge coloring, the adjacent strong edge coloring, the total coloring and the adjacent strong total coloring, the methods of the study and some unsolved problems. the D(β)- vertex-distinguishing edge-coloring generalizes the adjacent strong edge coloring. The vertex-distinguishing edge-coloring is special in the D(β)- vertex-distinguishing edge-coloring. So the study of the D(β)- vertex-distinguishing edge-coloring can make use of the method of researching the adjacent strong edge coloring.Secondly, this paper presents a general overview of the basic concepts and the research status quo of the vertex-distinguishing edge coloring and the vertex-distinguishing total coloring. The special graphs have some special structures and characteristics, so many researches of the problems of the graph theory are proceeded from them so that people can obtain the general results.We study some special graphs which have their adjacent strong edge chromatic numbers, mainly K₃^t , D_m,4, W_n , F_mâ–½S_n and F_mâ–½F_n , analysis the proving method and discuss their vertex-distinguishing edge coloring. We prove that the vertex-distinguishing edge-coloring conjecture holds for these graphs. On the basis of their vertex-distinguishing edge chromatic number, the paper discusses their vertex-distinguishing total coloring and gives the vertex-distinguishing edge chromatic numbers. Finally, after the conception of the vertex-distinguishing edge-coloring is proposed, only a few works have been done, because it's complex. This paper studys the D(β)- vertex-distinguishing edge-coloring and presents a general overview of the basic concepts and the research status quo of the D(β)- vertex-distinguishing edge coloring. This paper researches the D(β)- vertex-distinguishing edge coloring of some special graphs, C_m·P_n and C_m·F_n , mainly their D(2)- vertex-distinguishing edge coloring and vertex-distinguishing edge coloring in some situation, and proves that they hold for the correspondent conjectures. The paper investigates the method of researching the vertex-distinguishing edge coloring based on the way of studying the D(β)- vertex-distinguishing edge coloring.