On The Fractional Factor Problem Of Graphs

Graph theory is an old and young discipline. It is developing very rapidly the past two decades and used widely in a new branch of mathematics. There is a wide range of applications of graph theory in many fields, such as physics, chemistry, operations research, computer science, information theory, cybernetics all aspects of network the-ory, social science, economic management and other disciplines. Therefore, studying problems in graph theory is very important both in theory and in reality.Factor theory is one of the main problems in Graph Theory, which receives most research in the graph theory study. The study on the factors of graphs started over one hundred years ago, but until the 1970s did it become popular. So far, many results on the existence of factors in graphs have been gained.This thesis is divided into three chapters. The second and third chapter are the main contents of this thesis, in these two chapters, we obtain some results about frac-tional ID-[a, b]-factor-critical graphs and fractional (a, b, m)-deleted graphs.The first chapter is the basis of this paper. We first introduce some basic concep-tions,the progress of factors of graphs and fractional factors of graphs and the main results of this article.In Chapter 2, we investigate some sufficient conditions which guarantee the ex-istence of fractional ID-[a,b]-factor-critical graph. This chapter is divided into three parts. In the first section, we discuss the relationship between the binding numbers and the existence of critical graph. In the second section, we show the minimum de-gree and the neighborhoods of independent sets condition of critical graph. In the last section, we introduce the degree condition of critical graph.In Chapter 3, we study some sufficient conditions which guarantee the existence of fractional (a, b, m)-deleted graph. One part is to show the toughness condition of deleted graph. The of other is to discuss the binding numbers condition of deleted graph.