Current Graph Construction And Triangulation Embedding Of The Complete Graph K₁₆

In the process of embedding complete graphs into surfaces(orientable or nonorientable),the key point is to construct current graphs with different assignment modes satisfying the conditions.In this paper,we try to construct all current graphs of the complete graph K12s+4(s=1)by establishing and solving equations,and then study the number of orientable triangulation embeddings of different isomorphisms of the complete graph L12s+4(s=1).The whole paper is divided into four chapters.The arrangement and main contents of each chapter are as follows:The first chapter is the introduction,which mainly introduces the origin of graph theory,the current situation of the research on the embedding problem of complete graphs at home and abroad,and introduces the main framework of this paper.The second chapter is the current graph construction of K16.This chapter gives a method to find all the pairing ways of two currents through a system of column equations for the complete graph K16 current graph,which leads to all the current assignment ways,and thus to all the current graphs.Firstly,according to the assignment requirements of the first current,the current graph of the complete graph K16 is classified according to the different paths formed by the edges with the first current 1.Then,for each type of current graph,by establishing a set of equations,all the pairing modes satisfying the construction conditions are solved by computer programming.A set of solutions corresponds to a pairing mode,that is,a current assignment mode.The number of current assignment ways that can be calculated for the current graph of the complete graph K16 is eight.The third chapter is based on all current graphs of complete graph K16 obtained in the second chapter.According to the relevant theoretical knowledge in topological graph theory,the number of current graphs that are not strongly isomorphic in each pair of current graphs of all assignment modes is calculated to be 4 by classification and comparison,and the number of different triangulation embedding of complete graph K16 on the oriented surface is finally calculated to be 32.The fourth chapter summarizes the main research results of this paper,and points out some shortcomings of this paper and prospects for the future.