Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

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=2ap1a1p2a2⋯pkak(p1,p2,…,pk(p1,p2,…,pk are distinct odd primes and ai>0ai>0 for each i⩾1)i⩾1), there are t   distinct reflexible regular embeddings of the complete bipartite graph Kn,nKn,n up to isomorphism, where t=1t=1 if a=0a=0, t=2kt=2k if a=1a=1, t=2k+1t=2k+1 if a=2a=2, and t=3·2k+1t=3·2k+1 if a⩾3a⩾3. And, there are s   distinct self-Petrie dual regular embeddings of Kn,nKn,n up to isomorphism, where s=1s=1 if a=0a=0, s=2ks=2k if a=1a=1, s=2k+1s=2k+1 if a=2a=2, and s=2k+2s=2k+2 if a⩾3a⩾3.