{
  "id": "2106.00852",
  "version": "v1",
  "published": "2021-06-01T23:25:05.000Z",
  "updated": "2021-06-01T23:25:05.000Z",
  "title": "On the Cogirth of Binary Matroids",
  "authors": [
    "Cameron Crenshaw",
    "James Oxley"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The cogirth, $g^\\ast(M)$, of a matroid $M$ is the size of a smallest cocircuit of $M$. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that $g^\\ast(M)\\leq \\frac{1}{2}\\vert E(M)\\vert$ when $M$ is binary, unless $M$ simplifies to a projective geometry. We also show that, when equality holds, $M$ simplifies to a Bose-Burton geometry, that is, a matroid of the form $PG(r-1,2)-PG(k-1,2)$. These results extend to matroids representable over arbitrary finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T23:25:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "binary matroid",
      "arbitrary finite fields",
      "equality holds",
      "simplifies",
      "smallest cocircuit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}