The Adjacent-vertex-strongly-distinguishing E-total Coloring Of Some Graphs

Let G be a single graph.A total coloring f of G called adjacent-vertex-strongly-distinguishing E-total coloring, if no two adjacent vertex of G receive same color and no edge with incident vertex assigned the same color,and for any two adjacent vertexes u and v,there must be C(u)? C(v),where C(u) is a set of color of the vertex, and the color of the edge of the vertex is associated with,the color of the vertex adjacent to the vertex, The minimal number of colors required for the adjacent-vertex-strongly-distinguishing E-total coloring of G is called the adjacent-vertex-strongly-distinguishing E-total chromatic numbers.In this paper, according to the structure properties of some graphs,proof by con-tradiction, the structural patchwork method, constructed coloring function method and exhaustion method are used to study the adjacent-vertex-strongly-distinguishing E-total coloring of several basic graphs, some double graphs and some join graphs and some prod-uct graphs. Then the upper bond for the adjacent-vertex-strongly-distinguishing E-total chromatic number by the probability method was studied.The paper is divided into five parts:In the first part, we introduce some fundamental concepts and symbols which will be used throughout this paper.In the second part, we study the adjacent-vertex-strongly-distinguishing E-total coloring is given and some important results are introduced.In the third part, we discuss the adjacent-vertex-strongly-distinguishing E-total col-oring question of several double graphs and some join graphs.In the fourth part, we study the adjacent-vertex-strongly-distinguishing E-total col-oring question of the three product graphIn the fifth part, the upper bond for the adjacent-vertex-strongly-distinguishing E-total chromatic number by the probability method was studied.