Sachs triangulations and regular maps
type: Item
instance of: scholarly article
title: Sachs triangulations and regular maps
zbMATH Open document ID: 0814.05032
DOI: 10.1016/0012-365X(93)E0071-B
author: Jozef Širáň
author: Martin Škoviera
author: Heinz-Juergen Voss
published in: Discrete Mathematics
publication date: 1995-05-28T00:00:00Z
review text: The authors are interested in constructing ``highly symmetrical'' triangulations of closed surfaces. They note that known results on obtaining such triangulations are based on groups. The authors exploit an idea of Sachs concerning a method of triangulating surfaces. The adaptation of this idea is to define a Sachs triangulation of a closed surface \(S\) as a triangulation \(T\) with a vertex labeling \(\lambda\) in a group \(G\) so that: (1) For any facial triangle \(t\) of \(T\) with vertices \(x\), \(y\), \(z\), either \(\lambda(x) \lambda(y) \lambda(z)= 1\) or \(\lambda(x) \lambda(z) \lambda(y)= 1\). (2) For any \(g\), \(h\) in \(G\), there exists at most one edge in \(T\) with end points labelled \(g\) and \(h\). The authors consider the highest degree of symmetry to occur when the triangulation becomes a regular map. A triangulation is a regular map if for any two edges \(e\) and \(f\) with arbitrarily assigned directions there exists a map-automorphism sending \(e\) to \(f\) and preserving directions. In this paper, new proofs of fundamental results about Sachs triangulations are obtained, with focus on conditions under which they become regular maps. Also, a characterization of the label sets is given.
reviewed by: M.Marx
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C78
zbMATH DE Number: 701692
zbMATH Keywords: triangulation of closed surfaces
zbMATH Keywords: Sachs triangulation
zbMATH Keywords: vertex labeling
zbMATH Keywords: group
zbMATH Keywords: regular map
zbMATH Keywords: label sets
MaRDI profile type: Publication
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full work available at URL: 0012-365x(93)e0071-b
OpenAlex ID: W1973062853
description: scientific article; zbMATH DE number 701692