On the number of nonisomorphic orientable regular embeddings of complete graphs
type: Item
instance of: scholarly article
title: On the number of nonisomorphic orientable regular embeddings of complete graphs
zbMATH Open document ID: 1025.05017
author: Vladimir P. Korzhik
author: Heinz-Juergen Voss
published in: Journal of Combinatorial Theory. Series B
publication date: 2002-12-10T00:00:00Z
review text: In this paper those 2-cell orientable embeddings of a complete graph \(K_{n+1}\) are considered which are generated by rotation schemes on an abelian group \(\Phi\) of order \(n+1\), where a rotation scheme on \(\Phi\) is defined as a cyclic permutation \((\beta_1,\beta_2,\dots, \beta_n)\) of all nonzero elements of \(\Phi\). It is shown that two orientable embeddings of \(K_{n+1}\) generated by schemes \((\beta_1,\beta_2,\dots, \beta_n)\) and \((\gamma_1,\gamma_2,\dots, \gamma_n)\) are isomorphic if and only if \((\gamma_1, \gamma_2,\dots, \gamma_n)= (\varphi(\beta_1), \varphi(\beta_2),\dots, \varphi(\beta_n))\) or \((\gamma_1,\gamma_2,\dots, \gamma_n)= (\varphi(\beta_n),\dots, \varphi(\beta_2), \varphi(\beta_1))\), where \(\varphi\) is an automorphism of \(\Phi\). As a consequence, representing schemes by index one current graphs, the following results are obtained. For every \(s\geq 1\) the graphs \(K_{12s+4}\) and \(K_{12s+7}\) have no less than \(4^s\) mutually nonisomorphic orientable 3-face-colorable triangular embeddings. For every \(s\geq 1\) the graph \(K_{8s+5}\) has no less than \(14\cdot 8^{s-1}\) mutually nonisomorphic orientable quadrangular embeddings.
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 1843738
zbMATH Keywords: orientable surface
zbMATH Keywords: complete graph
zbMATH Keywords: orientable embeddings
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: jctb.2000.1993
OpenAlex ID: W2023344889
cites work: Exponential families of non-isomorphic triangulations of complete graphs
cites work: Q5782525
cites work: Face 2-colourable triangular embeddings of complete graphs
cites work: Surface embeddings of Steiner triple systems
cites work: Q3757929
cites work: Minimal quadrangulations of orientable surfaces
cites work: Q3766517
cites work: Three nonisomorphic triangulations of an orientable surface with the same complete graph
cites work: A sequence representation for maps
cites work: Uniqueness and faithfulness of embedding of toroidal graphs
cites work: Q4773723
DOI: 10.1006/JCTB.2000.1993
description: scientific article; zbMATH DE number 1843738