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

We prove a theorem that for an integer s⩾0s⩾0, if 12s+712s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of KnKn, where n=(12s+7)(6s+7)n=(12s+7)(6s+7), is at least 2n3/2(2/72+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 2αnℓ+o(nℓ)2αnℓ+o(nℓ), ℓ>1ℓ>1, nonisomorphic nonorientable triangular embeddings of KnKn for n=6t+1n=6t+1, t≡2mod3. 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.