{
  "id": "2203.12549",
  "version": "v1",
  "published": "2022-03-23T17:09:11.000Z",
  "updated": "2022-03-23T17:09:11.000Z",
  "title": "Double circuits in bicircular matroids",
  "authors": [
    "S. Guzmán-Pro",
    "W. Hochstättler"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of Hochst\\\"attler-Ne\\v{s}et\\v{r}il. The class of gammoids is a class of $M(K_4)$-free orientable matroids and it is the minimal minor-closed class that contains all transversal matroids. Towards proving the previous statement for the class of gammoids, Goddyn, Hochst\\\"attler, and Neudauer conjectured that every gammoid has a positive coline (equivalently, a positive double circuit), which implies that all orientations of gammoids are $3$-colourable. In this brief note we disprove Goddyn, Hochst\\\"attler, and Neudauers' conjecture by exhibiting a large class of bicircular matroids that do not contain positive double circuits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T17:09:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bicircular matroids",
      "first non-trivial case",
      "conjecture",
      "oriented matroids reads",
      "free oriented matroid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}