Total Coloring And Adjacent Vertex-distinguish 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. many issues in discrete systems and combinatorial analysis can be translated into the problem of graph coloring.For example, the edge maximum of n ? order graph which don't contain a certain graph G as a subgraph depends on the chromatic number of the graph. Therefore T R.Jensen and B.Toft asserted : the graph coloring theory in discrete mathematics at the center position. In real life, many areas will be dealt with the object of a certain set of rules according to certain classification of the problem, For example, schedule, scheduling, time-table problem, the problem of storing, circuit arrangement, task allocation, and so on. These problems are closely related to coloring theory, it is also the case, the practical application of graph coloring theories generated worldwide interest.The so-called graph coloring is refers speaking of the graph in the vertex,edge (to the plane graph also has face) and so on the element carries on the classification according to the certain rule. Object dissimilarity or rule dissimilarity, then has all kinds of colotings, after continues the vertex coloring,edge coloring as well as the face coloring of discrete map, the people also put forward the concept of total coloring.Such as vertex coloring,edge coloring,total coloring,strong coloring,adjacent strong edge coloring, adjacent vertex-distinguishing total coloring, vertex-distinguishing edge coloring ,edge-face coloring,perfect coloring and so on hundred and thousand of colouring ways.In order to properly express large-network, storage, the timing and allocation of research topics such as the relationship between the elements, Graph coloring theory staining as a viable tool for the introduction by the natural. Because of its good application background, Graph coloring theory has become the rapid development subcourse of the modern graph field.First,This paper presents a general overview of the basic concepts and research the status quo of the total coloring, then harmonization of the literature on the concept of coloring. Based on the theory, received total chromatic number of some graph, and through the total chromatic of these graph test to determine total chromatic conjecture(TCC). Reached some theorem of total chromatic, in the actual application of total chromatic, given a certain algorithm of total chromatic. Secondly, the article introducing adjacent vertex-distinguishing total coloring, adjacent vertex-distinguishing total coloring is the latest theory of a research direction, Zhang Zhong Fu others put forward the concept of adjacent vertex-distinguishing total coloring, some results have been given, and brought up the conjecture, Now,results little known that there are many unresolved issues, the adjacent vertex-distinguishing total coloring is the focus of this paper. given several types of graph adjacent vertex-distinguishing chromatic number, test and verify adjacent vertex-distinguishing chromatic conjecture.