The Embeddings Of Complete Graphs And Complete Bipartite Graphs On Surface

Symmetric graphs and symmetric maps are one of the classical fields of algebraic graph theory and topological graph theory.Especially the highly symmetric graphs and maps have always been one of the hot topics studied by many experts and scholars at home and abroad,because of their important value in theory and application.This dissertation mainly studies the embeddings of complete graphs and complete bipartite graphs on orientable surfaces.The content includes embeddability and embedded enumeration.The dissertation is consisted of seven chapters.The arrangement and main contents of each chapter are as follows:The first chapter is the introduction,which introduces the research background and current situation of complete bipartite maps and complete maps,and gives the main work and research ideas of this dissertation.The second chapter is the preliminary knowledge,which lists some basic knowledge and results of group theory,graph theory and maps.In Chapter 3,we study complete bipartite graphs with exactly two orientably edge transitive embeddings.By analyzing the structure of the automorphism group of the complete bipartite maps,we obtain the automorphism groups of the two orientably edge-transitive embeddings: one is the abelian exact bicyclic group,and the other is the non-abelian exact bicyclic group.Moreover,there must be an orientably edge-transitive embedding in the sense of isomorphism for the abelian exact bicyclic group.Next,the uniqueness of the non-abelian exact bicyclic group is given.At last,the complete classification of the complete bipartite graph with exactly two orientably edge-transitive embeddings is obtained.In Chapter 4,we study the enumeration of orientably edge-regular(that is edgetransitive but not arc-transitive)complete maps.By analyzing which automorphism subgroups of complete graphs are edge-transitive rather than arc-transitive,we obtain the automorphism group of orientably edge-regular complete maps.Next,the general structure of the orientable edge-transitive complete map is given,and a series of properties of the orientable edge-regular map are determined by analyzing the constructed map.Furthermore,the number of the different embeddings and non-isomorphic embeddings of orientably edge-regular complete maps is determined by combination method.In Chapter 5 and Chapter 6,we respectively study the counting problems of orientably vertex-primitive complete maps,and the counting problems of orientably vertex-transitive complete maps when vertex stabilizers of its automorphism groups are cyclic groups.By analyzing the different actions of the vertex stabilizers of the map automorphism group on the vertex set,we obtain the enumeration of different embeddings when the vertex stabilizers acting on the neighborhood of vertex have any number of orbits.Furthermore,in Chapter 5,we give the exact counting formula of non-isomorphic of orientably vertex-primitive complete maps when the number of orbits is prime.In Chapter 6,we get the maximum number of non-isomorphic of orientably vertex-transitive complete maps when the number of orbits is even.In Chapter 7,we sum up this dissertation and put forward its further prospects.