The Classification Of Double-face Complete Regular Dessins

Complete dessins are the 2-cell embeddings of complete bipartite graphs on the orientable surface,which are called regular if all edges are equivalent under the automorphism group of direction-preserving and point-preserving sets.The classification of complete regular dessins is one of the frontier topics in the research of algebraic graph theory and topological graph theory,which is an open problem proposed by the leader of graph theory-Gareth.Jones in 2007.Because the classification problem of complete regular dessins contains many complex situations,it is a huge project and a great challenge to complete this classification problem.In this thesis,we mainly study the classification of double-face complete regular dessins,which is a special case of the classification problem proposed by Jones,and also a generalization of the classification results of uniface regular dessins published by Fan and Li on Journal of Algebraic Combinatorics,which is of great significance in the algebraic curve.On the other hand,the classification of double-face complete regular dessins determines the structure of a class of bicyclic groups with index 2.Therefore,it is of great significance to study the classification of double-face complete regular dessins in algebraic combination,topological graph theory and group theory.This thesis mainly uses group theory and other methods to transform the classification problem of complete regular dessins into a group theory problem.