We prove a theorem that for an integer s ≥ 0, if 12 s + 7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of Kn, where n = (12 s + 7) (6 s + 7), is at least 2n3 / 2 (sqrt(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 2α nℓ + o (nℓ), ℓ > 1, nonisomorphic nonorientable triangular embeddings of Kn for n = 6 t + 1, t ≡ 2 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. © 2007 Elsevier B.V. All rights reserved.
Korzhik, V. P., & Kwak, J. H. (2008). A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs. Discrete Mathematics, 308(7), 1072–1079. https://doi.org/10.1016/j.disc.2007.03.060