On a Generalization of Spikes

Nick Brettell, Rutger Campbell, Deborah Chun, Kevin Grace, Geoff Whittle

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
124 Downloads (Pure)

Abstract

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.
Original languageEnglish
Pages (from-to)358-372
Number of pages15
JournalSIAM Journal on Discrete Mathematics
Volume33
Issue number1
DOIs
Publication statusPublished - 21 Feb 2019

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