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, Steve

N1 - Funding Information:
The first author’s work was supported by the UI&D Matemática e Aplicações of the University of Aveiro, through the Programa Operacional Ciência, Tecnologia, Inovaçao (POCTI) of the Fundação para a Ciência e a Tecnologia (FCT), and co-financed by the European Community fund FEDER. The second author’s work was supported by the Marsden Fund of New Zealand, project number UOA-721. The third author’s work was supported by the research grants NNSF(10971144) and BNSF(1092010) in China. The fourth author’s work was supported by a Korea Research Foundation grant, funded by the Korean Government (MOEHRD, Basic Research Promotion Fund), number KRF-2008-331-C00049. The fifth author’s work was supported by grant VEGA 1/0722/08 of the Slovak Republic’s Ministry of Education.

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

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U2 - 10.1007/s10801-010-0242-8

DO - 10.1007/s10801-010-0242-8

M3 - Article

AN - SCOPUS:79952445258

VL - 33

SP - 215

EP - 238

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 2

ER -