CLASSIFICATION OF REFLEXIBLE REGULAR EMBEDDINGS AND SELF-PETRIE DUAL REGULAR EMBEDDINGS OF COMPLETE BIPARTITE GRAPHS

Abstract

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say n = 2(a) p(1)(a1) p(2)(a2) ... p(k)(ak) (p(1,) p(2,)...,p(k) are distinct odd primes and a(i) > 0 foreach i >= 1), there are t distinct reflexible regular embeddings of the complete bipartite graph K-n,K-n up to isomorphism, where t = 1 if a = 0, t = 2(k) if a = 1, t = 2(k+1) if a = 2, and t = 3 . 2(k+1) if a >= 3. And, there are s distinct self-Petrie dual regular embeddings of K-n,K-n up to isomorphism, where s = 1 if a = 0, s = 2(k) if a = 1, s = 2(k+1) if a = 2, and s = 2(k+2) if a >= 3. (C) 2007 Elsevier B.V. All rights reserved.