On the minimal nonzero distance between triangular embeddings of a complete graph

Given two triangular embeddings f and f ′ of a complete graph K and given a bijection φ : V (K) → V (K), denote by M(φ) the set of faces (x, y, z) of f such that (φ(x), φ(y), φ(z)) is not a face of f . The distance between f and f ′ is the minimal value of |M(φ)| as φ ranges over all bijections between the vertices of K. We consider the minimal nonzero distance between two triangular embeddings f and f ′ of a complete graph. We show that 4 is the minimal nonzero distance in the case when f and f ′ are both nonorientable, and that 6 is the minimal nonzero distance in each of the cases when f and f ′ are orientable, and when f is orientable and f ′ is nonorientable. Running head: Distances between embeddings Corresponding author: M. J. Grannell, 17 Royal Avenue, Leyland, Lancashire, PR25 1BQ, England. Tel: 01772-433430 email: [email protected] AMS classifications: 05C10.