Unavoidable minors of large 3-connected matroids
type: Item
instance of: scholarly article
title: Unavoidable minors of large 3-connected matroids
zbMATH Open document ID: 0897.05020
author: Guoli Ding
author: Bogdan Oporowski
author: Dirk Vertigan
author: James Oxley
published in: Journal of Combinatorial Theory. Series B
publication date: 1998-10-05T00:00:00Z
review text: An earlier result of the same authors for 3-connected matroids is extended to the more general class of all 3-connected matroids. It is shown that for every integer \(n>2\) there exists a number \(N(n)\) such that every 3-connected matroid with at least \(N(n)\) elements has a minor isomorphic to an \((n+2)\)-point line or its dual, the cycle or cocycle matroid of \(K_{3,n}\), the cycle matroid of a wheel with \(n\) spokes, a whirl of rank \(n\), or a (uniform) \(n\)-spike. Introducing the idea of a hamiltonian partial representation of matroids which have a spanning circuit and using the fact that, up to duality, every sufficiently large 3-connected matroid has a big 3-connected hamiltonian minor, the proof of this theorem can be built on the earlier results for 3-connected binary matroids based on their matrix representations and some Ramsey-theoretic results for matrices. Furthermore it is shown that the above-mentioned result is best possible in the sense that the given list of matroids contains no redundancy.
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 1154574
zbMATH Keywords: matroid
zbMATH Keywords: Ramsey theory
zbMATH Keywords: unavoidable minor
reviewed by: Kathrin Klamroth
MaRDI profile type: Publication
OpenAlex ID: W2068984122
cites work: Unavoidable minors of large 3-connected binary matroids
cites work: Q3997075
cites work: Typical subgraphs of 3- and 4-connected graphs
cites work: Q3135082
cites work: Ramsey numbers for matroids
cites work: Connectivity in Matroids
DOI: 10.1006/JCTB.1997.1785
description: scientific article; zbMATH DE number 1154574