Classification of regular embeddings of n-dimensional cubes

  • Authors:

  • Domenico A. Catalano;Marston D. Conder;Shao Fei Du;Young Soo Kwon;Roman Nedela;Steve Wilson

  • Affiliations:

  • Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal 3810-193;Department of Mathematics, University of Auckland, Auckland, New Zealand;School of Mathematical Sciences, Capital Normal University, Beijing, China 100048;Department of Mathematics, Yeungnam University, Kyongsan, Republic of Korea 712-749;Mathematical Institute, Slovak Academy of Sciences, Banská Bystrica, Slovakia 975 49;Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, USA 86011

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2011

Quantified Score

Hi-index 0.00

Visualization

Abstract

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 驴 of order 1 or 2 in the symmetric group S n such that 驴 fixes n, preserves the set of all pairs B i ={i,i+m} for 1驴i驴m, and induces the same permutation on this set as the permutation B i 驴 B f(i) for some additive bijection f:驴 m 驴驴 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.