A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs

Abstract

We prove a theorem that for an integer $s⩾0$, if $12s+7$ is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of $K_n$, where $n=(12s+7)(6s+7)$, is at least $2^{n^{3/2}(\sqrt{2}/72+\mathrm{o}(1))}$. By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least $2^{\alpha n^{\ell }+\mathrm{o}(n^{\ell })}$, $\ell >1$, nonisomorphic nonorientable triangular embeddings of $K_n$ for $n=6t+1$, $t\equiv 2\mathrm{mod}3$. To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given.