{
  "abstract": "For a graph $G$, let $\\mathrm{tw}(G)$, $\\mathrm{tree}\\text{-}α(G)$, and $\\mathrm{tree}\\text{-}χ(G)$ denote its treewidth, tree-independence number, and tree-chromatic number, respectively. We prove that every nonempty graph $G$ satisfies $\\mathrm{tw}(G)+1\\leq\\mathrm{tree}\\text{-}α(G)\\mathrm{tree}\\text{-}χ(G)(1+\\lfloor\\log_2(\\mathrm{tree}\\text{-}α(G))\\rfloor)$. This improves the quadratic bound of Koutsoutis, Krause, Liu, Redzic, and Ueckerdt. For every pair of integers $n,k\\geq2$, we construct a graph $G$ with $\\mathrm{tree}\\text{-}α(G)=n$, $\\mathrm{tree}\\text{-}χ(G)=k$, and $\\mathrm{tw}(G)=Ω(nk\\log_2 n)$. Thus the upper bound is best possible up to a constant factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20407",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinheng Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n4dzs6ct1fvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20407",
  "fulltext_url": null,
  "id": "2n4dzs6ct1fvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20407",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222136055",
  "pdf_url": "https://arxiv.org/pdf/2610.20407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n4dzs6ct1fvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Treewidth and the Product of Tree-Independence and Tree-Chromatic Numbers",
  "title_text": "Treewidth and the Product of Tree-Independence and Tree-Chromatic Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/2n4dzs6ct1fvy",
  "venue": "arXiv",
  "versions": []
}