Cyclic degrees of 3-polytopes
type: Item
instance of: scholarly article
title: Cyclic degrees of 3-polytopes
zbMATH Open document ID: 0930.05055
author: Oleg V. Borodin
author: Douglas R. Woodall
published in: Graphs and Combinatorics
publication date: 2000-01-30T00:00:00Z
review text: If \(G=(V,E,F)\) is a 3-polytope or a 3-connected plane graph, and \(v\in V\), then the cyclic degree \(d_{c}(v)\) is the number of other vertices that share incident faces with \(v\) and \(\Delta ^{*}(G)\) denotes the maximum face degree in \(G\). In this paper it is proved that if \(k\geq 3\) and \(G\) is a 3-polytope with \(\Delta ^{*}(G)\leq k\), then \(G\) has a vertex \(v\) with \(d(v)\leq 5\) and \(d_{c}(v)\leq M(k)\), where \(M(k)\) is given in the paper and is best possible for every \(k\geq 3\). Note that \(M(k)=k+3\) for \(17\leq k\leq 20\) and \(M(k)= k+2\) for every \(k\geq 21\). This implies improvements to some known upper bounds for the cyclic chromatic numbers of 3-polytopes.
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 1369700
zbMATH Keywords: cyclic degree
zbMATH Keywords: 3-polytope
zbMATH Keywords: cyclic chromatic number
zbMATH Keywords: 3-connected plane graph
zbMATH Keywords: face degree
zbMATH Keywords: truncated dodecahedron
reviewed by: Ioan Tomescu
MaRDI profile type: Publication
full work available at URL: s003730050060
OpenAlex ID: W2003216770
DOI: 10.1007/S003730050060
description: scientific article; zbMATH DE number 1369700