On Smarandachely Adjacent Vertex Total Coloring Of Several Graphs

Let G be a finite and undirected simple graph with vertex set V(G) and edge setE(G), an normal total coloring of G is called a smarandachely adjacent vertex total coloring if|C(u)\C(v)|≥1且|C(v)\C(u)|≥1 for(?)uv<E E(G), whereC(u) is a set of color of the color ofu together with edge incident withu. The required minimal number of colors in coloring method is denotedχsat(G).In this paper, according to the structure properties of some graphs, the structural patchwork method, constructed coloring function method and exhaustion method are used to study the Smarandachely adjacent vertex total coloring of several corona graphs and two classes of join graph, two classes of kMulti-Mycielski graphs, some product graphs and two classes of 3-regular graphs.The paper is divided into five parts:In the first part, we introduce some fundamental concepts and symbol which will be used throughout this paper.In the second part, we study the Smarandachely adjacent vertex total coloring ques-tion of several corona graphs and two classes of join graphs, and give their Smarandachely adjacent vertex Total chromatic numbers.In the third part, we discuss the Smarandachely adjacent vertex total coloring ques-tion of two classes of kMulti-Mycielski graphs, and give their Smarandachely adjacent vertex Total chromatic numbers.In the fourth part, we study the Smarandachely adjacent vertex total coloring ques-tion of the three product graph by operating cycles and paths,cycles and give their S-marandachely adjacent vertex total chromatic numbers.In the fifth part, we construct two kinds of 3-regular graphs Rk,m and G(V, E), give their corresponding chromatic numbers of Smarandachely adjacent vertex total, verify the conclusion which the two classes graphs satisfy to conjecture of Smarandachely adjacent vertex total coloring at the same time.