Classification of regular embeddings of hypercubes of odd dimension (Q861801)

A regular imbedding of a graph \(G\) into a closed surface is a 2-cell imbedding whose automorphism group acts regularly on flags; that group will then have order \(4|E(G)|\). Kwon and Nedela [Discrete Math, to appear] showed that the \(n\)-cube \(Q_n\) has no nonorientable regular imbeddings, if \(n> 2\). \textit{R. Nedela} and \textit{M. Skoviera} [Eur. J. Comb. 18, 807--823 (1997; Zbl 0908.05036)] constructed regular orientable imbeddings for \(Q_n\), for each solution of \(e^2\equiv 1\pmod n\), showed that different solutions give nonisomorphic maps, and conjectured that there are no other regular imbeddings of \(Q_n\). The present authors affirm this conjecture, for all odd \(n\).