Smarandachely Adjacent Vertex Distinguishing E-Total Coloring Of Some Graphs

For a simple graph G,f is called an E-Total coloring of G, which means that adjacent vertices of G dyed different colors and the incident elements of G dyed different colors. If the color sets of two adjacent vertices are different,f is called adjacent vertex distinguishing E-Total coloring. Furthermore, if the color sets of two adjacent vertices don’t contain each other,f is called Smarandachely adjacent vertex distinguishing E-Total coloring and the minimal number of colors, which the Smarandachely adjacent vertex E-total coloring required, is called the Smarandachely adjacent vertex E-total chromatic number. While the color set of the vertex is made up of the color of the vertex and the colors of its incident edges.In recent years, the mainly basic methods of studying the graph coloring are exhaustion method, structural patchwork method and composition analysis method. The so-called exhaustive method means searching for a colored diagram from the isomorphism class of this graph to explain the existence of the chromatic number when the upper or lower boundary of the chromatic number of some coloring method is known. The structural patchwork method refers to color the partial graph firstly, then color the entire graph through a patchwork. composition analysis method refers to the analysis of the structure of graph, and the use of the combination of knowledge of graph coloring. This paper mainly applies the above-mentioned three methods to study the Smarandachely adjacent vertex distinguishing E-Total coloring of three product graph, which are consisted by the path, circle, star, fan, wheel(Direct product, Cartesian product, lexicographic product), some join graph, some Corona graphs as well as some kinds of3-regular graphs, and get their Smarandachely adjacent vertex distinguishing E-Total chromatic number. the conjecture of Smarandachely adjacent vertex distinguishing E-total coloring is effective in these graphs.This paper is divided into four chapters:The first chapter introduces some basic concepts and theory of graph coloring and Smarandachely adjacent vertex distinguishing E-total coloring.The second chapter studies the Smarandachely adjacent vertex distinguishing E-Total coloring of the simple graph (path, circle, star, fan, wheel and complete graph), the join graph and Corona graphs of these simple graphs, gets their Smarandachely adjacent vertex distinguishing E-Total chromatic number and verify the conjecture of Smarandachely adjacent vertex distinguishing E-total coloring.Third chapter mainly studies the Smarandachely adjacent vertex distinguishing E-Total coloring of the three product graph (Direct product, Cartesian product, as well as lexicographic product) between path, circle, star, fan, wheel, gets their Smarandachely adjacent vertex distinguishing E-Total chromatic number and further verify the conjecture of Smarandachely adjacent vertex distinguishing E-total coloring.The fourth chapter first constructs two kinds of3-regular graphs, studies the Smarandachely adjacent vertex distinguishing E-Total coloring of these two kinds of3-regular graph, and gets the Smarandachely adjacent vertex distinguishing E-Total chromatic number of generalized3-regular ring graph and a class of generalized Petersen graph, further verify the conjecture of Smarandachely adjacent vertex distinguishing E-total coloring is effective in these graphs.