{
  "abstract": "The first open case of Hadwiger's conjecture states that every $K_7$-minor-free graph is 6-colorable. We prove that this is the case for $K_7^=$-minor-free graphs, where $K_7^=$ denotes the graph obtained from $K_7$ by deleting two independent edges. The proof is based on an independently interesting density result: Every 5-connected $K_7^=$-minor-free graph with $n\\ge 6$ vertices has at most $4n-8$ edges.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17760",
  "author_total": 3,
  "authors": [
    {
      "name": "Zdeněk Dvořák"
    },
    {
      "name": "Sergey Norin"
    },
    {
      "name": "Neil Rahman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ph3en3ae25kz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17760",
  "fulltext_url": null,
  "id": "9ph3en3ae25kz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17760",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ph3en3ae25kz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Every graph with no $K_7^=$ minor is 6-colorable",
  "title_text": "Every graph with no K_7^= minor is 6-colorable",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/9ph3en3ae25kz",
  "venue": "arXiv",
  "versions": []
}