Regular embeddings of canonical double coverings of graphs
type: Item
instance of: scholarly article
title: Regular embeddings of canonical double coverings of graphs
zbMATH Open document ID: 0856.05029
author: Roman Nedela
author: Martin Škoviera
published in: Journal of Combinatorial Theory. Series B
publication date: 1997-02-17T00:00:00Z
review text: This paper addresses the question of determining, for a given graph \(G\), all regular maps having \(G\) as their underlying graph, i.e., all embeddings of \(G\) in closed surfaces exhibiting the highest possible symmetry. We show that if \(G\) satisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor product \(G \otimes K_2\), can be described in terms of regular embeddings of \(G\). This allows us to ``lift'' the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the ``derived'' maps by employing those of the ``base'' maps. We apply these results to determining all orientable regular embeddings of the tensor products \(K_n \otimes K_2\) (known as the cocktail-party graphs) and of the \(n\)-dipoles \(D_n\), the graphs consisting of two vertices and \(n\) parallel edges joining them. In the first case we show, in particular, that regular embeddings of \(K_n \otimes K_2\) exist only if \(n\) is a prime power \(p^l\), and there are \(2 \phi (n - 1)\) or \(\phi (n - 1)\) isomorphism classes of such maps (where \(\phi\) is Euler's function) according to whether \(l\) is even or odd. For \(l\) even an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings of \(D_n\) exist for each positive integer \(n\), and their number is a power of 2 depending on the decomposition of \(n\) into primes.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 934814
zbMATH Keywords: automorphism
zbMATH Keywords: regular maps
zbMATH Keywords: embeddings
zbMATH Keywords: closed surfaces
zbMATH Keywords: canonical double covering
zbMATH Keywords: tensor product
zbMATH Keywords: regular embeddings
MaRDI profile type: Publication
full work available at URL: jctb.1996.0044
OpenAlex ID: W2073668070
DOI: 10.1006/JCTB.1996.0044
Recommended article: On even-closed regular embeddings of graphs
Recommended article: Q3424896
Recommended article: Regular embeddings of complete multipartite graphs
Recommended article: Regular embeddings of complete bipartite graphs
Recommended article: Regular orientable imbeddings of complete graphs
Recommended article: Graphs with Cayley canonical double covers
Recommended article: Classification of Regular Embeddings of Complete Multipartite Graphs
Recommended article: Regular embeddings of complete bipartite graphs: classification and enumeration
Recommended article: Regular embeddings of cycles with multiple edges revisited
Recommended article: Regular orientable embeddings of complete bipartite graphs
description: scientific article; zbMATH DE number 934814