### 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 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≤ i≤ m, 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.

Original language | English (US) |
---|---|

Pages (from-to) | 215-238 |

Number of pages | 24 |

Journal | Journal of Algebraic Combinatorics |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2011 |

### Fingerprint

### Keywords

- Chiral
- Cubes
- Hypercubes
- Regular embeddings
- Regular maps

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Algebra and Number Theory

### Cite this

*Journal of Algebraic Combinatorics*,

*33*(2), 215-238. https://doi.org/10.1007/s10801-010-0242-8

**Classification of regular embeddings of n-dimensional cubes.** / Catalano, Domenico A.; Conder, Marston D E; Du, Shao Fei; Kwon, Young Soo; Nedela, Roman; Wilson, Stephen E.

Research output: Contribution to journal › Article

*Journal of Algebraic Combinatorics*, vol. 33, no. 2, pp. 215-238. https://doi.org/10.1007/s10801-010-0242-8

}

TY - JOUR

T1 - Classification of regular embeddings of n-dimensional cubes

AU - Catalano, Domenico A.

AU - Conder, Marston D E

AU - Du, Shao Fei

AU - Kwon, Young Soo

AU - Nedela, Roman

AU - Wilson, Stephen E

PY - 2011/3

Y1 - 2011/3

N2 - 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≤ i≤ m, and induces the same permutation on this set as the permutation B i → B f(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.

AB - 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≤ i≤ m, and induces the same permutation on this set as the permutation B i → B f(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.

KW - Chiral

KW - Cubes

KW - Hypercubes

KW - Regular embeddings

KW - Regular maps

UR - http://www.scopus.com/inward/record.url?scp=79952445258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952445258&partnerID=8YFLogxK

U2 - 10.1007/s10801-010-0242-8

DO - 10.1007/s10801-010-0242-8

M3 - Article

VL - 33

SP - 215

EP - 238

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 2

ER -