An improved bound for the crossing number of \(C_m\times C_n\): a self-contained proof using mostly combinatorial arguments
type: Item
instance of: scholarly article
title: An improved bound for the crossing number of \(C_m\times C_n\): a self-contained proof using mostly combinatorial arguments
zbMATH Open document ID: 1053.05037
author: Gelasio Salazar
author: Edgardo Ugalde
published in: Graphs and Combinatorics
publication date: 2004-12-13T00:00:00Z
review text: Let \(C_n\) denote the \(n\)-cycle graph, and let \(\times \) denote the Cartesian product of graphs. In 1973 Harary, Kainen, and Schwenk conjectured that the crossing number of \(C_n\times C_m\) for \(n\geq m\geq 3\) is \(n(m-2)\). This conjecture was recently proved if \(n\geq m(m+1)\), i.e. finitely many possible exceptions are left for every \(m\), see [\textit{L. Y. Glebsky} and \textit{G. Salazar}, J. Graph Theory 47, 53--72 (2004; Zbl 1053.05032)]. The current paper shows that for all \(m,n\) sufficiently large, almost 80 crossings are there.
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 2121763
zbMATH Keywords: crossing number
reviewed by: László A. Székely
MaRDI profile type: Publication
full work available at URL: s00373-003-0549-5
OpenAlex ID: W2004255417
DOI: 10.1007/S00373-003-0549-5
description: scientific article; zbMATH DE number 2121763