{
  "abstract": "We prove that every triangle-free graph with minimum degree at least $\\frac{n}{3}$ is $4$-colorable and thereby settle a problem of Brandt and Thomassé (2005) at the threshold $\\frac{n}{3}$. The number four is best possible. For a positive integer-valued function $f(n)=o(n)$, we relate the chromatic number of $f(n)$-vertex subgraphs of the Kneser graph $KG(n,f(n))$ to that of triangle-free graphs with minimum degree at least $\\frac{n}{3}-f(n)$. Consequently, for every $0<δ<1$ and $\\varepsilon>0$, and for all sufficiently large $n$, every $n$-vertex triangle-free graph with minimum degree at least $\\frac{n}{3}-n^{1-δ}$ has chromatic number at most $10^{391}+1+\\left\\lceil{(1+\\varepsilon)(1-δ)}/δ\\right\\rceil$. We also show that every sufficiently large $n$-vertex maximal triangle-free graph with minimum degree at least $\\frac{n}{3}-f(n)$ and chromatic number at least $10^{391}$ contains a bipartite subgraph with parts of orders $\\frac{n}{3}-O(f(n))$ and $\\frac{2n}{3}-O(f(n))$; the remaining induced subgraph admits a homomorphism to $KG(\\frac{n}{3}-O(f(n)),O(f(n)))$. Finally, we connect maximal $K_r$-free graphs with minimum degree at least $\\frac{2r-5}{2r-3}n-f(n)$ to $K_{r-1}$-free graphs and extend these results to $K_r$-free graphs. Our proofs employ the recent strong Brandt--Thomassé theorem of Łuczak, Polcyn, and Reiher.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24097",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiaao Li"
    },
    {
      "name": "Xinyuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45tvpf9fp9aae.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24097",
  "fulltext_url": null,
  "id": "45tvpf9fp9aae",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24097",
  "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": "W7214054225",
  "pdf_url": "https://arxiv.org/pdf/2609.24097",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45tvpf9fp9aae.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24097",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24097"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounded chromatic number of graphs with small clique number and large minimum degree",
  "title_text": "Bounded chromatic number of graphs with small clique number and large minimum degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/45tvpf9fp9aae",
  "venue": "arXiv",
  "versions": []
}