{
  "abstract": "A graph is $H$-free if it has no induced subgraph isomorphic to $H$. In 2020, Chudnovsky, Goedgebeur, Schaudt, and Zhong characterized all graphs $H$ such that there are only finitely many minimal obstructions to $3$-colouring $H$-free graphs. In general, the minimal obstructions to $k$-colouring $H$-free graphs are the $(k+1)$-vertex-critical $H$-free graphs, those are, the $H$-free graphs $G$ with $χ(G)=k+1$ but $χ(G-v)=k$ for every vertex in $G$. In this paper we complete the characterization for all $k > 4$ by showing that there are onky finitely $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+\\ell P_1$ for some $\\ell \\geq 0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10737",
  "author_total": 3,
  "authors": [
    {
      "name": "Iain Beaton"
    },
    {
      "name": "Ben Cameron"
    },
    {
      "name": "Adam van Omme"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fsabajyh3xeqn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10737",
  "fulltext_url": null,
  "id": "fsabajyh3xeqn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10737",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221087966",
  "pdf_url": "https://arxiv.org/pdf/2610.10737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fsabajyh3xeqn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Obstructions to $k$-colouring $H$-free graphs",
  "title_text": "Obstructions to k-colouring H-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/fsabajyh3xeqn",
  "venue": "arXiv",
  "versions": []
}