Exponentially many genus embeddings of the complete graph K12s+3

In this paper, we consider the problem of construction of exponentially many distinct genus embeddings of complete graphs. There are three approaches to solve the problem. The first approach is to construct exponentially many current graphs by the theory of graceful labellings of paths; the second approach is to find a current assignment of the current graph by the theory of current graph; the third approach is to find exponentially many embedding (or rotation) scheme of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph. According to these three approaches, we can construct exponentially many distinct genus embeddings of complete graph K12s+3, which show that there are at least \(\frac{1}{2} \times {\left( {\frac{{200}}{9}} \right)^s}\) distinct genus embeddings for K12s+3.