On The Dmarandachely Adjacent Vertex Distinguishing Total Coloring Of Several Kinds Graph

Let G be a simple graph, a total coloring f of G is called Smarandachely-adjacent vertex total coloring if no two adjacent vertices of G receive the same color and no two adjacent edges of G are dyed same color, what's more, for any two adjacent ver-tices u and v, there must be|C(u)\C(v)|≥1 and|C(v)\C(u)|≥1. And the minimal number of colors required for the Smarandachely-adjacent vertex total color-ing of G is called the Smarandachely-adjacent vertex total chromatic number, where C(u)={f(u)}{∪{f(uv)|uv∈E(G)}.Recently, the exhaustive method, the combination of analytic method and the Pieced method of constructing are three main methods of researching graphs. The exhaustive method mainly through coloring every elements of the graph which is selected from its isomorphic category to state the chromatic number exist, at same times, this coloring bound of the isomorphic category of the graph has been known; the combination of an-alytic method mainly apply the combinational knowledge to study the coloring through analyse the construction of the graph; the combination of constructing use the coloring of its parts to combine the whole graph. In this paper, based on above methods, the Smarandachely-adjacent vertex total coloring of path, cycle, star, fan, wheel, complete graph, some incident graphs, four kinds of products(Direct product, cartesian product, strong product, semi-strong product) and some 3-regular graphs are discussed, and the Smarandachely-adjacent vertex total chromatic number of path, cycle, star, fan, wheel, complete graph, some incident graphs, four kinds of products (Direct product, cartesian product, strong product, semi-strong product) and some 3-regular graphs are obtained.There are four parts in this paper:In the first part, some fundamental concepts and conjectures of graphs are introduced.In the second part, definition of the Smarandachely-adjacent vertex total coloring is given and some simple graphs and incident graphs are obtained.In the third part, the Smarandachely-adjacent vertex total chromatic number of four products (Direct product, cartesian product, strong product, semi-strong product) are given by building the Smarandachely-adjacent vertex total coloring function. Moreover, the conjecture of Smarandachely-adjacent vertex total coloring of graphs is checked.In the fourth part, some kinds of 3-regular graphs are constructed, and the Smara-ndachely-adjacent vertex total chromatic number of those graphs are given through building the Smarandachely-adjacent vertex total coloring function. Furthermore, the conjecture of Smarandachely-adjacent vertex total coloring of 3-regular graphs is checked.