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 a 1 p 2 a 2 ⋯ p k a k ( p 1 , p 2 , … , p k are distinct odd primes and a i > 0 for each i ⩾ 1 ) , there are t distinct reflexible regular embeddings of the complete bipartite graph K n , 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 , 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 .