Adjacent Vertex Strongly Distinguishing E-total Coloring Of Some Graphs

Let G(V,E) is a simple connected graph, k is a positive integer.if a mapping of f from:E(G)?V(G) to {1,2, … , k} satisfies the condition (?)uv ? E(G), f(u)? f(V), f(v) ? f(uv), f(u)?f(uv). The (?)uv ? E(G),C(u) ? C(v), where C(u) = f(u) ? f(v) ? f(uv)|uv ? E(G). then f is called an adjacent strong vertex distinguishing E-total coloring of graph, which the repuired minimal number of k is called the adjacent strong vertex distinguishing E-total chromatic number of G.In this paper, The distribution of the overall color method, proof by contradiction, the analysis of composite structure method and the constructing coloring function method are used to study the adjacent vertex strong distinguishing E-total coloring question of several direct product graphs,some corona and join graphs, path and cycle of classes of Graphs.On the basis of the above, We obtain the corresponding chromatic numbers of them.Then two upper bond for the adjacent vertex strong distinguishing E-total chromatic number by the new probability method was studied.This paper is divided into five parts:In the first part, we introduce some fundamental concepts which will be used throughout some important results are introduced in this paper.In the second part, we discuss the adjacent vertex strongly distinguishing E-total coloring question of direct product, strong product, lexi-cographic product, semistrong product and then give the total chromatic number.In the thrid part, we discuss the adjacent vertex strongly distinguishing E-total coloring of a class of graphs of some corona and join graphs and and then give the total chromatic number.In the thrid part, we discuss the adjacent vertex strongly distinguishing E-total coloring of a class of graphs of path and cycle and then give the total chromatic number.In the fourth part, we discuss the strong adjacent vertex distinguishing E-total coloring of the graph of the circle and the circle, and give its total chromatic number.In the fifth part, two upper bond for the adjacent vertex strong distinguishing E-total chro-matic number by the new probability method was studied.