Biembeddings of metacyclic groups and triangulations of orientable surfaces by complete graphs
type: Item
instance of: scholarly article
title: Biembeddings of metacyclic groups and triangulations of orientable surfaces by complete graphs
zbMATH Open document ID: 1253.05047
author: Michael John Grannell
author: Martin Knor
published in: The Electronic Journal of Combinatorics
publication date: 2012-10-24T00:00:00Z
full work available at URL: v19i3p29
review text: Summary: For each integer \(n \geq 3\), \(n \neq 4\), for each odd integer \(m \geq 3\), and for any \(\lambda \in {\mathbb{Z}}_n\) of (multiplicative) order \(m'\) where \(m' \mid m\), we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group \({\mathbb{Z}}_m\ltimes_{\lambda} {\mathbb{Z}}_n\). This extends the spectrum of Latin squares known to be biembeddable.The best existing lower bounds for the number of triangular embeddings of a complete graph \(K_z\) in an orientable surface are of the form \(z^{z^2(a-o(1))}\) for suitable positive constants \(a\) and for restricted infinite classes of \(z\). Using embeddings of \({\mathbb{Z}}_3 \ltimes_{\lambda} {\mathbb{Z}}_n\), we extend this lower bound to a substantially larger class of values of \(z\).
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 6098371
zbMATH Keywords: triangular embedding
zbMATH Keywords: Latin square
zbMATH Keywords: complete graph
zbMATH Keywords: complete tripartite graph
zbMATH Keywords: metacyclic group
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6098371