{
  "abstract": "Erdős et al. (JCT-B, 1989) conjectured that, for integers $r\\ge 2$ and $δ\\ge 2$ with $3r-1\\midδ$, every connected $K_{2r+1}$-free graph of order $n$ and minimum degree $δ$ has diameter at most $ \\frac{3r-1}{r}\\cdot \\frac{n}δ+O(1)$. Czabarka et al. (JCT-B, 2021) later proposed the following generalization: for every $k\\ge 3$ and $δ\\ge\\left\\lceil\\frac{3k}{2}\\right\\rceil-1$, every connected $K_{k+1}$-free graph of order $n$ and minimum degree at least $δ$ has diameter at most $(3-\\frac{2}{k})\\cdot\\frac{n}δ+O(1)$. We disprove the latter conjecture, including its $k$-colorable version, for every $k\\ge 7$ and sufficiently large $δ$. When $k=2r\\ge 8$ and $3r-1\\midδ$, our construction also disproves the conjecture of Erdős et al. (JCT-B, 1989).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03346",
  "author_total": 2,
  "authors": [
    {
      "name": "Hangdi Chen"
    },
    {
      "name": "Yaojun Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/532ps97jbxyzw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03346",
  "fulltext_url": null,
  "id": "532ps97jbxyzw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03346",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7208739590",
  "pdf_url": "https://arxiv.org/pdf/2609.03346",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/532ps97jbxyzw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03346",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03346"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counterexamples to two conjectures on the diameter of clique-free graphs",
  "title_text": "Counterexamples to two conjectures on the diameter of clique-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/532ps97jbxyzw",
  "venue": "arXiv",
  "versions": []
}