On Edge Domination Number Of Graphs

Graph theory was originated in 1736, with Euler solved the problem of Konigsberg seven bridges as a milestone. After nearly three hundred years of development, graph theory constantly keeps growing, and forms many new branches. But the domination theory of graphs was formally put forward until half a century ago. In 1958, Claude Berge first gave the concept of domination number. Four years later, Oystein Ore defined the domination set and the domination number. T.W. Haynes, S.T. Hedetniemi and P.J. Slater published two monographs about domination theory of graphs in 1998. Then the domination theory began to rapidly develop. It has both the scientific theoretical value and the application value. Due to the time of the development of domination theory is relatively short, various theories are not perfect. In order to better enrich and perfect the domination theory of graphs, we continue to deeply research.This article is mainly going to study several aspects as follows.In the first chapter of this article, we describe the origin of the graph theory, the developing situation of the domination theory, some basic concepts of theory graphs, and the main research work of this paper.The second chapter mainly introduces some upper or lower bounds of signed domination numbers and signed edge domination numbers about graphs, with some signed domination numbers and signed edge domination numbers of special graphs.In the third chapter, we research the signed cycle domination number of the cartesian product graph, gain the lower bounds of the signed cycle domination number about the cartesian product graph, and put forward a new domination function of edge balance domination function. We conduct a preliminary study on it.The fourth chapter of this article, we study the edge-domatic number of graphs, obtain the edge-domatic number and the total edge-domatic number of some special graphs.The fifth chapter discusses the fractional domination and the total fractional domination,gives the fractional domination numbers and the total fractional domination numbers of some special graphs.We sum up the results what we have obtained in the last chapter, and point out the future research directions in the dominating theory.