{
  "id": "2307.01184",
  "version": "v1",
  "published": "2023-07-03T17:50:07.000Z",
  "updated": "2023-07-03T17:50:07.000Z",
  "title": "Finding dense minors using average degree",
  "authors": [
    "Kevin Hendrey",
    "Sergey Norin",
    "Raphael Steiner",
    "Jérémie Turcotte"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\\sqrt{2}-1-o(1))\\binom{t}{2}$ edges. We show that this cannot be improved beyond $\\left(\\frac{3}{4}+o(1)\\right)\\binom{t}{2}$. Finally, for $t\\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-03T17:50:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C35",
      "05C83"
    ],
    "keywords": [
      "average degree",
      "finding dense minors",
      "vertex minor",
      "hadwigers conjecture",
      "exactly determine"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}