{
  "abstract": "Let $\\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\\ge7$ vertices with at least $4n-9$ edges contains a $\\Ke$ minor, settling Conjecture~1.4 of Dvo\\v rák, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stronger statement that every $4$-bilight graph on $n\\ge4$ vertices with at least $4n-9$ edges contains either a $\\Ke$ minor or a $K_6$ subgraph. Within their reduction framework, we strengthen the rooted-minor theorem. We show that every $4$-light 5-rooted graph of rooted $4$-density at least two has a model with two nonroot vertices and at most one missing edge incident with them. At the critical density, reductions preserve density exactly, which prevents them from creating a new $K_6$ subgraph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26041",
  "author_total": 4,
  "authors": [
    {
      "name": "Caibing Chang"
    },
    {
      "name": "Zijian Deng"
    },
    {
      "name": "Qinfei Tang"
    },
    {
      "name": "Caihong Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52atntnbs3f4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26041",
  "fulltext_url": null,
  "id": "52atntnbs3f4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26041",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214073234",
  "pdf_url": "https://arxiv.org/pdf/2609.26041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52atntnbs3f4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A sharp density bound for 5-connected graphs with no $\\Ke$ minor",
  "title_text": "A sharp density bound for 5-connected graphs with no Ke minor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/52atntnbs3f4p",
  "venue": "arXiv",
  "versions": []
}