Small cocircuits in minimally vertically $4$-connected matroids
| Autor: | Oxley, James, Walsh, Zach |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific non-binary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit. Comment: There was an error in the proof of Theorem 1.4. We removed the proof, and replaced Theorem 1.4 with Conjecture 1.6 |
| Databáze: | arXiv |
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