Classification of regular embeddings of n-dimensional cubes

Title
Classification of regular embeddings of n-dimensional cubes
Author(s)
권영수Domenico A. Catalano[Domenico A. Catalano]Marston D.E. Conder[Marston D.E. Conder]Shao Fei Du[Shao Fei Du]Roman Nedela[Roman Nedela]Steve Wilson[Steve Wilson]
Keywords
COMPLETE BIPARTITE GRAPHS; MAPS; HYPERMAPS; POWER; ODD
Issue Date
201103
Publisher
SPRINGER
Citation
JOURNAL OF ALGEBRAIC COMBINATORICS, v.33, no.2, pp.215 - 238
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 in 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 sigma of order 1 or 2 in the symmetric group S-n such that sigma 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 bar right arrow 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.
URI
http://hdl.handle.net/YU.REPOSITORY/25517http://dx.doi.org/10.1007/s10801-010-0242-8
ISSN
0925-9899
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이과대학 > 수학과 > Articles
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