{
  "id": "2505.10097",
  "version": "v1",
  "published": "2025-05-15T08:58:13.000Z",
  "updated": "2025-05-15T08:58:13.000Z",
  "title": "Odd Hadwiger's conjecture for the complements of Kneser graphs",
  "authors": [
    "Meirun Chen",
    "Reza Naserasr",
    "Lujia Wang",
    "Sanming Zhou"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A generalization of the four-color theorem, Hadwiger's conjecture is considered as one of the most important and challenging problems in graph theory, and odd Hadwiger's conjecture is a strengthening of Hadwiger's conjecture by way of signed graphs. In this paper, we prove that odd Hadwiger's conjecture is true for the complements $\\overline{K}(n,k)$ of the Kneser graphs $K(n,k)$, where $n\\geq 2k \\ge 4$. This improves a result of G. Xu and S. Zhou (2017) which states that Hadwiger's conjecture is true for this family of graphs. Moreover, we prove that $\\overline{K}(n,k)$ contains a 1-shallow complete minor of a special type with order no less than the chromatic number $\\chi(\\overline{K}(n,k))$, and in the case when $7 \\le 2k+1 \\le n \\le 3k-1$ the gap between the odd Hadwiger number and chromatic number of $\\overline{K}(n,k)$ is $\\Omega(1.5^{k})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-15T08:58:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C22",
      "05C83"
    ],
    "keywords": [
      "odd hadwigers conjecture",
      "kneser graphs",
      "complements",
      "chromatic number",
      "odd hadwiger number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}