Unavoidable Minors of Matroids with Minimum Cocircuit Size Four
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916840266006528 |
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| author | Mizell, Matthew Oxley, James |
| author_facet | Mizell, Matthew Oxley, James |
| contents | In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which every cocircuit has size at least four has $F_7, M^*(K_{3,3}), M(K_5),$ or $ M(K_{2,2,2})$ as a minor. Generalizing these results, this paper proves that every simple matroid in which all cocircuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unavoidable Minors of Matroids with Minimum Cocircuit Size Four Mizell, Matthew Oxley, James Combinatorics In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which every cocircuit has size at least four has $F_7, M^*(K_{3,3}), M(K_5),$ or $ M(K_{2,2,2})$ as a minor. Generalizing these results, this paper proves that every simple matroid in which all cocircuits have at least four elements has as a minor one of nine matroids, seven of which are well known. All nine of these special matroids have rank at most five and have at most twelve elements. |
| title | Unavoidable Minors of Matroids with Minimum Cocircuit Size Four |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.09015 |