On The Number Of Triangular Embeddings Of Some Complete Graphs

The minimum genus of a graph G,denoted by γ(G),is the minimum genusγ(S)of orientable surfaces S in which G has a 2-cell embedding.The determination of the minimum genus of complete graphs,was divided into twelves cases,i.e.we need to calculate the minimum genus of K12s+i,for i = 0,1,…11.It is known that a complete graph Kn,has an orientable triangular embedding if and only if n =0,3,4,7(mod12).According to graceful labellings and current graph theory of the path graph,Luis Goddyn,et al,proved there are at least 11s non-isomorphic oriented triangular embeddings of K12s+7.According to current graph theory,Vladimir P.Korzhik showed there are at least 4s non-isomorphic oriented triangular embeddings of K12s+4.In this paper,we improve the lower bound for the number of triangular embeddings of K12s+4.In Chapter 1,we introduce the number of minimum genus embeddings of complete graphs,the background as well as some basic concepts needed in the thesis.At the same time,we also introduces the basic structure of this paper.In Chapter 2,we introduce some results and conclusions.In Chapter 3,we find a new current labeling method for the current graph of K12s+-7,and the current graph of K12s+4 is constructed by double graceful labeling of P2s+1,and the number of double graceful labelings of P2s+1 is estimated.In Chapter 4,we find the number of one-rotation systems for the current graph-s.Furthermore,the lower bound of the number of oriented triangular embeddings of 12s+4 is obtained.