On the number of triangular embeddings of some complete graphs

College of Mathematics and Econometrics, HuNan University, ChangSha 410000

Abstract： It is known that a complete graph Kn, has an orientable triangular embedding if and only if n ≡ 0, 3, 4, 7(mod12). In this paper, we study the number of non-isomorphic oriented triangular embeddings of K12s+4 和K12s+7. 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. In this paper, we find a new current labeling method for the current graph 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, according to double graceful labeling of path graph and the number of one-rotation systems for the current graph, we improve the lower bound for the number of triangular embeddings of K12s+4.

Keywords： embedding minimum genus current graph K12s+ 7