Classification of regular embeddings of hypercubes of odd dimension
type: Item
instance of: scholarly article
title: Classification of regular embeddings of hypercubes of odd dimension
zbMATH Open document ID: 1112.05027
author: Shaofei Du
author: Roman Nedela
author: Jin Ho Kwak
published in: Discrete Mathematics
publication date: 2007-02-02T00:00:00Z
review text: A regular imbedding of a graph \(G\) into a closed surface is a 2-cell imbedding whose automorphism group acts regularly on flags; that group will then have order \(4|E(G)|\). Kwon and Nedela [Discrete Math, to appear] showed that the \(n\)-cube \(Q_n\) has no nonorientable regular imbeddings, if \(n> 2\). \textit{R. Nedela} and \textit{M. Skoviera} [Eur. J. Comb. 18, 807--823 (1997; Zbl 0908.05036)] constructed regular orientable imbeddings for \(Q_n\), for each solution of \(e^2\equiv 1\pmod n\), showed that different solutions give nonisomorphic maps, and conjectured that there are no other regular imbeddings of \(Q_n\). The present authors affirm this conjecture, for all odd \(n\).
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 5121355
zbMATH Keywords: regular map
zbMATH Keywords: genus
zbMATH Keywords: arc-transitive graph
zbMATH Keywords: permutation group
reviewed by: Arthur T. White
MaRDI profile type: Publication
full work available at URL: j.disc.2006.05.035
OpenAlex ID: W2019628954
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DOI: 10.1016/J.DISC.2006.05.035
description: scientific article; zbMATH DE number 5121355