On size, circumference and circuit removal in 3-connected matroids
type: Item
instance of: scholarly article
title: On size, circumference and circuit removal in 3-connected matroids
zbMATH Open document ID: 0951.05020
DOI: 10.1016/S0012-365X(99)00379-9
author: Manoel Lemos
author: James Oxley
published in: Discrete Mathematics
publication date: 2000-12-13T00:00:00Z
review text: In this paper several extremal results for 3-connected matroids are proven. The results are best possible which is shown by examples attaining the bounds. More specific it is shown that for a 3-connected matroid \(M\): (i) If the rank \(r(M)\) is at least 6 then the circumference \(c(M)\) is at least 6 (Theorem 1.5, having a very ingenious proof) and if moreover \(|E(m)|\geq 4r(M)- 5\) then there is a circuit that when deleted from \(M\) leaves a 3-connected matroid (Corollary 4.1). (ii) If \(r(M)> 4\) and \(M\) has a basis \(B\) such that \(M\setminus e\) is not 3-connected for all \(e\) in \(E(M)-B\) then \(|E(M)|\leq 3r(M)- 4\). (Theorem 1.2 which is a refinement of Theorem 1.1. The proof of Theorem 1.2 heavily relies on Theorem 1.1, which is proven in a separate paper.) (iii) If \(M\) is minimally 3-connected but not Hamiltonian then \(|E(M)|\leq 3r(M)- C(M)\) (Theorem 1.4 sharpening a previous result by Oxley).
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 1462280
zbMATH Keywords: extremal results
zbMATH Keywords: 3-connected matroids
zbMATH Keywords: circumference
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1462280