The 3-connected binary matroids with circumference 6 or 7
type: Item
instance of: scholarly article
title: The 3-connected binary matroids with circumference 6 or 7
zbMATH Open document ID: 1181.05021
author: Raul Cordovil
author: Bráulio jun. Maia
author: Manoel Lemos
published in: European Journal of Combinatorics
publication date: 2009-12-03T00:00:00Z
review text: For a matroid \(M\) that has a circuit, \(\text{circ}(M)\) denotes the circumference of \(M\), that is, the maximum cardinality of a circuit of \(M\). Recently, the circumference of a matroid has appeared in some bounds; [see \textit{M. Lemos} and \textit{J. Oxley}, Discrete Math. 189, No.\,1-3, 283-289 (1998; Zbl 0956.05029); J. Graph Theory 30, No.\,1, 51--66 (1999; Zbl 0933.05086); Discrete Math. 220, No.\,1-3, 145--157 (2000; Zbl 0951.05020)]. The results of this paper are used in \textit{R. Cordovil, B. M. Junior,} and \textit{M. Lemos} [Discrete Math. 309, No.\,4, 655--665 (2009; Zbl 1169.05012)] to improve a lower bound of \textit{M. Lemos} and \textit{J. Oxley} [Discrete Math. 220, No.,1-3, 145--157 (2000; Zbl 0951.05020] for the size of a 3-connected binary matroid having a circuit whose deletion originates also a 3-connected matroid. In this paper, the authors construct all 3-connected binary matroids with circumference equal to 6 or 7 having large rank. Moreover, they conjecture that the following problems are very difficult: {\parindent=5mm \begin{itemize}\item[1.]constructing all 3-connected matroids with circumference 6 or 7 (and large rank), \item[2.]constructing all the 3-connected binary matroids with circumference 8. \end{itemize}}
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 5641706
zbMATH Keywords: 3-connected matroid
zbMATH Keywords: binary matroid
zbMATH Keywords: circumference of a matroid
reviewed by: Kelly J. Pearson
MaRDI profile type: Publication
full work available at URL: j.ejc.2008.12.001
OpenAlex ID: W2145167242
cites work: Q3135082
cites work: On packing minors into connected matroids
cites work: On removable circuits in graphs and matroids
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cites work: Matroids having small circumference
cites work: Connected matroids with a small circumference
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DOI: 10.1016/J.EJC.2008.12.001
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description: scientific article; zbMATH DE number 5641706