Exponentially many nonisomorphic orientable triangular embeddings of \(K_{12s+3}\)
type: Item
instance of: scholarly article
title: Exponentially many nonisomorphic orientable triangular embeddings of \(K_{12s+3}\)
zbMATH Open document ID: 1167.05020
author: Vladimir P. Korzhik
published in: Discrete Mathematics
publication date: 2009-04-09T00:00:00Z
review text: The orientable case \(n\equiv\pmod{12}\) of the Heawood Map Color Theorem was solved by using, for each \(s\), an index three current graph with current group \(Z_{12s+3}\) (see \textit{G. Ringel} [''Map Color Theorem,'' Springer-Verlag, Berlin (1974; Zbl 0287.05102)]), to give an orientable triangular imbedding of \(K_{12s+3}\). The present author provides a new index three current graph, with the same current group and the same result. He then shows that, for \(s\geq 11\), there are at least \(2^{2s-11}\) nonisomorphic orientable triangular imbeddings of \(K_{12s+3}\), thus completing the proof that there are constants \(M\), \(c> 0\), and \(b\geq 1/12\) such that for every \(n\geq M\) there are at least \(c2^{bn}\) nonisomorphic orientable as well as nonorientable genus imbeddings of the complete graph \(K_n\).
reviewed by: Arthur T. White
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 5542380
zbMATH Keywords: topological embedding
zbMATH Keywords: nonisomorphic triangular embedding
zbMATH Keywords: complete graph
zbMATH Keywords: orientable surface
zbMATH Keywords: index three current graph
zbMATH Keywords: current group
MaRDI profile type: Publication
full work available at URL: j.disc.2008.01.027
OpenAlex ID: W2045645894
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DOI: 10.1016/J.DISC.2008.01.027
description: scientific article; zbMATH DE number 5542380