Researches Of Some Graph Embedding On Surfaces With Small Genus

How to imbed a graph into surfaces is a very important branch of the topological theory. It is Computing the number of imbeddings on surfaces with different genus that is genus distributions or total genus distributions of a graph. For the practicality in theroy field,attracting many scholars’ attentions. As is known to all, imbedding distribution is NP-Complete problem. For many graphs, we haven’t known its imbedding distributions and total imbedding distributions yet. However, there is always relation among the imbeddings on surfaces of different genus. Therefore, it has significance to the study of imbeddings on some surfaces. In particular, it is important for investigating on the imbedding on sphere, torus, projective plane, Klein bottle. In this thesis, we research on the imbeddings of some graphs on surfaces with small genus, especially on projective plane.Recently, Many new results have been known by using the imbedding model of joint tree which professor Yanpei Liu introduced. In the following, we will introduce the content of each chapter briefly.In Chapter 1, we firstly state the concept of surface, imbedding and how to represent surfaces by polygons. Then we introduce the very important results and theory system of graph imbedding. In the following, the background of this thesis is presented.In Chapter 2, we introduce some important lemmas and theorems associ-ated with this article, such as the imbedding model of joint tree, the polygonal representations of projective plane and so on.In Chapter 3, we study on the embedding number of circular graph Hn on the torus and Klein bottle.In Chapter 4, we study on the embedding number of circular graph pn□pn on the projective plane.In Chapter 5, we firstly summarize the results of this thesis. Then we prospect the future research work.