Exponentially many maximum genus embeddings and genus embeddings for complete graphs

There are many results on the maximum genus, among which most are written for the existence of values of such embeddings, and few attention has been paid to the estimation of such embeddings and their applications. In this paper we study the number of maximum genus embeddings for a graph and find an exponential lower bound for such numbers. Our results show that in general case, a simple connected graph has exponentially many distinct maximum genus embeddings. In particular, a connected cubic graph G of order n always has at least distinct maximum genus embeddings, where α and m denote, respectively, the number of inner vertices and odd components of an optimal tree T. What surprise us most is that such two extremal embeddings (i.e., the maximum genus embeddings and the genus embeddings) are sometimes closely related with each other. In fact, as applications, we show that for a sufficient large natural number n, there are at least many genus embeddings for complete graph K _n with n ≡ 4, 7, 10 (mod12), where C is a constance depending on the value of n of residue 12. These results improve the bounds obtained by Korzhik and Voss and the methods used here are much simpler and straight. This work was supported by the National Natural Science Foundation of China (Grant No. 10671073), Science and Technology commission of Shanghai Municipality (Grant No. 07XD14011) and Shanghai Leading Academic Discipline Project (Grant No. B407)