{
  "abstract": "The binding number of a graph is the smallest ratio $|N(S)|/|S|$ among all nonempty vertex sets $S$ whose neighborhood $N(S)$ is not the whole vertex set. We prove that every finite nonempty $K_k$-free graph has binding number at most $k-2$ for every $k\\ge6$, confirming Brandt's conjecture; the bound is sharp for complete $(k-1)$-partite graphs with equal nonempty parts. We first prove the case $k=6$ and then extend it to all $k\\ge6$ by induction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28536",
  "author_total": 2,
  "authors": [
    {
      "name": "Jifu Lin"
    },
    {
      "name": "Heng Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/257khffyeg46k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28536",
  "fulltext_url": null,
  "id": "257khffyeg46k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28536",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223116079",
  "pdf_url": "https://arxiv.org/pdf/2610.28536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/257khffyeg46k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.28536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A sharp bound on the binding number of $K_k$-free graphs and the Brandt conjecture",
  "title_text": "A sharp bound on the binding number of K_k-free graphs and the Brandt conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/257khffyeg46k",
  "venue": "arXiv",
  "versions": []
}