Classification of regular embeddings of <Emphasis Type="Italic">n</Emphasis>-dimensional cubes |
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Authors: | Domenico A Catalano Marston D E Conder Shao Fei Du Young Soo Kwon Roman Nedela Steve Wilson |
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Institution: | 1.Departamento de Matemática,Universidade de Aveiro,Aveiro,Portugal;2.Department of Mathematics,University of Auckland,Auckland,New Zealand;3.School of Mathematical Sciences,Capital Normal University,Beijing,China;4.Department of Mathematics,Yeungnam University,Kyongsan,Republic of Korea;5.Mathematical Institute,Slovak Academy of Sciences,Banská Bystrica,Slovakia;6.Department of Mathematics and Statistics,Northern Arizona University,Flagstaff,USA |
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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. |
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Keywords: | |
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