{
  "id": "2110.13120",
  "version": "v2",
  "published": "2021-10-25T17:24:25.000Z",
  "updated": "2022-05-25T23:48:03.000Z",
  "title": "Small cocircuits in minimally vertically $4$-connected matroids",
  "authors": [
    "James Oxley",
    "Zach Walsh"
  ],
  "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",
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-05-25T23:48:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "connected matroid",
      "small cocircuits",
      "element cocircuit",
      "element matroid",
      "specific non-binary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}