On the maximal distance between triangular embeddings of a complete graph

• Authors: Vladimir P. Korzhik
• Venue: Journal of Combinatorial Theory Series B
• Year: 2006

The distance d(f, f') between two triangular embeddings f and f' of a complete graph is the minimal number t such that we can replace t faces in f by t new faces to obtain a triangular embedding isomorphic to f'. We consider the problem of determining the maximum value of d(f, f') as f and f' range over all triangular embeddings of a complete graph. The following theorem is proved: for every integer s ≥ 9, if 4s + 1 is prime and 2 is a primitive root modulo (4s + 1), then there are nonorientable triangular embeddings f and f of K12s+4 such that d(f, f') ≥ (1/2)(4s + 1)(12s + 4) - O(s), where (4s + 1)(12s + 4) is the number of faces in a triangular embedding of K12s+4. Some number-theoretical arguments are advanced that there may be an infinite number of odd integers s satisfying the hypothesis of the theorem.