Lower Bound of the Number of Maximum Genus Embeddings and Genus Embeddings of K _12s+7

Abstract

In this paper, we study lower bound of the number of maximum orientable genus embeddings (or MGE in short) for a loopless graph. We show that a connected loopless graph of order n has at least ${\frac{1}{4^{\gamma_M(G)}}\prod_{v\in{V(G)}}{(d(v)-1)!}}$ distinct MGE’s, where γ _M (G) is the maximum orientable genus of G. Infinitely many examples show that this bound is sharp (i.e., best possible) for some types of graphs. Compared with a lower bound of Stahl (Eur J Combin 13:119–126, 1991) which concerns upper-embeddable graphs (i.e., embedded graphs with at most two facial walks), this result is more general and effective in the case of (sparse) graphs permitting relative small-degree vertices. We also obtain a similar formula for maximum nonorientable genus embeddings for general graphs. If we apply our orientable results to the current graph G _s of K _12s+7, then G _s has at least 2^3s distinct MGE’s.This implies that K _12s+7 has at least (22)^s nonisomorphic cyclic oriented triangular embeddings for sufficient large s.