Regular embeddings of complete bipartite graphs (Q1850076)

Complete graphs are the only major class of graphs for which the complete (full) classification of regular embeddings on an orientable surface has been accomplished. For complete bipartite graphs and \(n\)-cube graphs a variety of embeddings have been constructed by \textit{R. Nedela} and \textit{M. Škoviera} [Eur. J. Comb. 18, 807-823 (1977; Zbl 0908.05036)]. Of these the best known is the embedding of the complete bipartite graph \(K_{n,n}\) with \(n\) faces, each bounded by a Hamiltonian circuit; see \textit{N. L. Biggs} and \textit{A. T. White} [Permutation groups and combinatorial structures (London Mathematical Society Lecture Notes Series 33, Cambridge University Press, Cambridge etc.) (1979; Zbl 0415.05002)]. In the paper under review the authors prove that, up to isomorphism, this is the only regular embedding of \(K_{p,p}\), \(p\) prime, on an orientable surface.