Journal of Algebraic Combinatorics

, Volume 33, Issue 2, pp 215-238

First online:

Classification of regular embeddings of n-dimensional cubes

  • Domenico A. CatalanoAffiliated withDepartamento de Matemática, Universidade de Aveiro
  • , Marston D. E. ConderAffiliated withDepartment of Mathematics, University of Auckland Email author 
  • , Shao Fei DuAffiliated withSchool of Mathematical Sciences, Capital Normal University
  • , Young Soo KwonAffiliated withDepartment of Mathematics, Yeungnam University
  • , Roman NedelaAffiliated withMathematical Institute, Slovak Academy of Sciences
  • , Steve WilsonAffiliated withDepartment of Mathematics and Statistics, Northern Arizona University

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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 Q n 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 Q n 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 S n such that σ fixes n, preserves the set of all pairs B i ={i,i+m} for 1≤im, and induces the same permutation on this set as the permutation B i B f(i) for some additive bijection f:ℤ m →ℤ m . 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.


Hypercubes Cubes Regular maps Regular embeddings Chiral