By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes Qn into nonorientable surfaces exist for any positive integer n > 2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807-823] presented a construction giving for each solution of the congruence e2 ≡ 1 (mod n) a regular embedding Me of the hypercube Qn into an orientable surface. It was conjectured that all regular embeddings of Qn into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes Qn into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n. © 2006 Elsevier B.V. All rights reserved.
Du, S. F., Kwak, J. H., & Nedela, R. (2007). Classification of regular embeddings of hypercubes of odd dimension. Discrete Mathematics, 307(1), 119–124. https://doi.org/10.1016/j.disc.2006.05.035