An infinite series of surfaces with known 1-chromatic number
type: Item
instance of: scholarly article
title: An infinite series of surfaces with known 1-chromatic number
zbMATH Open document ID: 0888.05023
author: Vladimir P. Korzhik
published in: Journal of Combinatorial Theory. Series B
publication date: 1997-09-15T00:00:00Z
review text: The 1-chromatic number \(\chi_1(S)\) of a surface \(S\) is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. It is proved that if \(4n+3\) is a prime number \((n\geq 0)\) then \(\chi_1(N_{8(2n+1)^2})= R(N_{8(2n+1)^2})\), where \(R(S)=\lfloor \frac12 (9+\sqrt {81-32E(S))}\rfloor\) is Ringel's upper bound for \(\chi_1(S)\), \(E(S)\) is the Euler characteristic of \(S\) and \(N_{8(2n+1)^2}\) is the nonorientable surface of genus \(8(2n+1)^2\). By Dirichlet's theorem the arithmetic progression \(4n+3\) \((n=1,2,3,\dots)\) contains an infinite number of prime integers. As a result the first known infinite series of surfaces with known 1-chromatic number is obtained.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C15
zbMATH DE Number: 1060753
zbMATH Keywords: graph embedding
zbMATH Keywords: graph immersion
zbMATH Keywords: surface
zbMATH Keywords: chromatic number
zbMATH Keywords: genus
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: jctb.1997.1792
OpenAlex ID: W2166027310
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DOI: 10.1006/JCTB.1997.1792
description: scientific article; zbMATH DE number 1060753