Enumerating Regular Graph Covering And Genera Of Cayley Maps

The regular coverings of graphs is one of important tools and methods in both of algebraic graph theory and topological graph theory.Recently,it has been used to some classifications and constructions for symmetric graphs and symmetric maps.Since Hofmeister counted the r-fold coverings of connected graphs in 1988,the prob-lem of enumerating on regular coverings of graphs attracts many famous researchers’attention.On the other hand,the numeration and genus distribution of maps are one of mainly research fields in topological graph theory.Many researchers have obtained plenty of results about this problems.Based on this,we will focus on several top-ics related to enumeration on graph regular coverings whose transformation group is a given group,genus distribution especially Cayley genus distribution and directed em-bedding,classification of regular t-balanced Cayley maps over a given group.The study in this area is currently one of the hot topics in the cross discipline of the three different branches of mathematics:group theory,graph theory and surface theory.We want to keep in step with the latest development in this field and to solve some related open problems.The thesis is organized as follows.Chapter 1 is an introduction part,where the first section gives a brief introduc-tion on the background of graph regular coverings and maps enumeration,the second section introduces the related concepts and useful lemmas and the third section is the organization of the thesis.In Chapter 2,the thesis classified Z2-extension of any cyclic group in the begin-ning.Then,by the enumerating formula of graph regular coverings,we enumerated the regular graph coverings whose transformation groups are Z2-extensions of cyclic group Z2n-1.Furthermore,we enumerated the regular graph coverings whose transformation groups are Z2-extensions of any cyclic group.In the last,we considered the enumer-ation on regular graph coverings whose transformation group is generalized dihedral group or generalized dicyclic group.In Chapter 3,Zp-extensions of a cyclic group with p odd prime were classified.Then we enumerated the regular graph coverings whose transformation groups are Zp-extensions of a cyclic group with p odd prime.In Chapter 4,firstly a formula of genus of Cayley map was given and then using this formula,we determined the Cayley genus polynomials of some classes of famous graphs in the network.In the first section,we got the Cayley genus polynomial of star graphs,bubble sort graphs and hypercubes.In the second section,we obtained the Cayley genus polynomial of alternating group network and in the third section,we determined the Cayley genus polynomial of the multi-dimensional torus.In Chapter 5,based on the notations and properties of directed embedding,Steiner triple system and current graphs and some useful lemmas,we used the current graph method to show that there exists a tournament,which can directed embedded in the orientable surface whose genus is[(n-3)(n-4)/12],on n vertices if and only if n≡ 3 or 7(mod 12).This partly answered the problem which is given by Bonnington et al.[J.Combin.Theory Ser.B,2002(85):1-20]that which tournaments on n vertices can be directed embedded into the orientable surface whose genus is[(n-3)(n-4)/12],which is equal to the genus of Kn.In Chapter 6,based on the properties of regular Cayley maps and some known results,we obtained the partial classification of the regular t-balanced Cayley maps on the direct product of two dihedral groups.In the last chapter,we summarized the results of the thesis and gave some problems to be further studied.