Matroids with the circuit cover property
type: Item
instance of: scholarly article
title: Matroids with the circuit cover property
zbMATH Open document ID: 0920.05016
author: Xudong Fu
author: Luis Goddyn
published in: European Journal of Combinatorics
publication date: 1999-09-10T00:00:00Z
full work available at URL: http://summit.sfu.ca/item/4787
review text: Let \(M\) be a binary matroid and \(p: M \mapsto Z^+\) be a weight on \(M\). A family \(\mathcal F\) of circuits of \(M\) is called a faithful circuit cover of \(M\) with respect to \(p\) if every element \(e\) of \(M\) is contained in precisely \(p(e)\) members of \(\mathcal F\). The following condition for \(p\), called the admissibility condition, is an obvious necessary condition for a matroid to have a faithful cover with respect to \(p\): (1) for every cocircuit \(D\) of \(M\), the total weight of \(D\) must be even; (2) for every cocircuit \(D\) and every \(e \in D\), \(p(e) \leq p(D)/2\). The faithful circuit cover theorem by \textit{B. Alspach, L. Goddyn} and \textit{C.-Q. Zhang} [Trans. Am. Math. Soc. 344, No. 1, 131-154 (1994; Zbl 0810.05043)] for graphs is generalized in this paper to binary matroids. That is, a matroid \(M\) has a faithful circuit cover with respect every admissible weight \(p\) if it does not have a minor isomorphic to any of \(F_7^*\), \(R_{10}\), \(M^*(K_5)\) or \(M(P_{10})\).
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1251349
zbMATH Keywords: faithful circuit cover
zbMATH Keywords: binary matroid
zbMATH Keywords: admissibility condition
MaRDI profile type: Publication
OpenAlex ID: W2122776236
DOI: 10.1006/EUJC.1998.0246
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description: scientific article; zbMATH DE number 1251349