Generalized spikes with circuits and cocircuits of different cardinalities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866918250761158656 |
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| author | Brettell, Nick Grace, Kevin |
| author_facet | Brettell, Nick Grace, Kevin |
| contents | We consider matroids with the property that every subset of the ground set of size $s$ is contained in a $2s$-element circuit and every subset of size $t$ is contained in a $2t$-element cocircuit. We say that such a matroid has the \emph{$(s,2s,t,2t)$-property}. A matroid is an \emph{$(s,t)$-spike} if there is a partition of the ground set into pairs such that the union of any $s$ pairs is a circuit and the union of any $t$ pairs is a cocircuit. Our main result is that all sufficiently large matroids with the $(s,2s,t,2t)$-property are $(s,t)$-spikes, generalizing a 2019 result that proved the case where $s=t$. We also present some properties of $(s,t)$-spikes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14524 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalized spikes with circuits and cocircuits of different cardinalities Brettell, Nick Grace, Kevin Combinatorics 05B35 We consider matroids with the property that every subset of the ground set of size $s$ is contained in a $2s$-element circuit and every subset of size $t$ is contained in a $2t$-element cocircuit. We say that such a matroid has the \emph{$(s,2s,t,2t)$-property}. A matroid is an \emph{$(s,t)$-spike} if there is a partition of the ground set into pairs such that the union of any $s$ pairs is a circuit and the union of any $t$ pairs is a cocircuit. Our main result is that all sufficiently large matroids with the $(s,2s,t,2t)$-property are $(s,t)$-spikes, generalizing a 2019 result that proved the case where $s=t$. We also present some properties of $(s,t)$-spikes. |
| title | Generalized spikes with circuits and cocircuits of different cardinalities |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2209.14524 |