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

We prove a theorem that for an integer image, if image is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of image, where image, is at least image. 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 image, image, nonisomorphic nonorientable triangular embeddings of image for image, image. 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.