{
  "abstract": "Let $P_t$ denote the induced path on $t$ vertices. Let $ω(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $χ(G)\\le(t-1)^{ω(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $χ(G)\\le (t-2)^{ω(G)-1}$ for $t\\ge4$. We lower the base of the exponential by one: for every $t\\ge5$, every $P_t$-free graph $G$ satisfies \\[ χ(G)\\le 3\\,(t-3)^{ω(G)+4}. \\] The proof combines two refinements of the Gyárfás path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08847",
  "author_total": 1,
  "authors": [
    {
      "name": "Sang-il Oum"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z7apty3v6wv4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08847",
  "fulltext_url": null,
  "id": "7z7apty3v6wv4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08847",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212087266",
  "pdf_url": "https://arxiv.org/pdf/2609.08847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z7apty3v6wv4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.08847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Coloring graphs with no long induced path",
  "title_text": "Coloring graphs with no long induced path",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/7z7apty3v6wv4",
  "venue": "arXiv",
  "versions": []
}