Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 215–238

Classification of regular embeddings of n-dimensional cubes


  • Domenico A. Catalano
    • Departamento de MatemáticaUniversidade de Aveiro
    • Department of MathematicsUniversity of Auckland
  • Shao Fei Du
    • School of Mathematical SciencesCapital Normal University
  • Young Soo Kwon
    • Department of MathematicsYeungnam University
  • Roman Nedela
    • Mathematical InstituteSlovak Academy of Sciences
  • Steve Wilson
    • Department of Mathematics and StatisticsNorthern Arizona University

DOI: 10.1007/s10801-010-0242-8

Cite this article as:
Catalano, D.A., Conder, M.D.E., Du, S.F. et al. J Algebr Comb (2011) 33: 215. doi:10.1007/s10801-010-0242-8


An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Qn were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n=2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Qn for all n. In particular, we show that for all even n (=2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group Sn such that σ fixes n, preserves the set of all pairs Bi={i,i+m} for 1≤im, and induces the same permutation on this set as the permutation BiBf(i) for some additive bijection f:ℤm→ℤm. We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n.


HypercubesCubesRegular mapsRegular embeddingsChiral

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© Springer Science+Business Media, LLC 2010