A map on an orientable surface is called regular if its automorphism group acts regularly on its dart set. It is called reversible if it is isomorphic (as a map) to its mirror image. The Petrie dual of a map \(M\) has the same underlying graph as \(M\) and the boundaries of its faces are Petrie polygons, that is zig-zag walks in \(M\).
Now let us quote the authors’ 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 \(n= 2^a p^{a_1}_1 p^{a_2}_2\cdots p^{a_k}_k\) (\(p_1,p_2,\dots, p_k\) are distinct odd primes and \(a_i> 0\) for each \(i\geq 1\)), 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 \(t= 3\cdot 2^{k+1}\) if \(a\geq 3\). 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 \(a\geq 3\).”