{
  "abstract": "For a graph $G$ and an integer $d \\geq 0$, let $χ^d(G)$ denote the $d$-defective chromatic number, and let $G \\boxtimes K_{d+1}$ be the $(d+1)$-fold clique blowup of $G$. Norin and Steiner disproved the conjecture $χ(G) = χ^d(G \\boxtimes K_{d+1})$ of Guo, Kang and Zwaneveld by exhibiting, for infinitely many $d$, graphs with $χ(G) \\geq (30/29) χ^d(G \\boxtimes K_{d+1})$, and they proved the universal upper bound $χ(G) \\leq 2 χ^d(G \\boxtimes K_{d+1})$. Writing $C_d = \\sup_G χ(G)/χ^d(G \\boxtimes K_{d+1})$ and $C^* = \\sup_d C_d$, their results give $C^* \\in [30/29, 2]$. We improve the lower bound: we exhibit an explicit 40-vertex graph $W$ with $χ(W) = 11$ and $χ^2(W \\boxtimes K_3) = 10$, so that $C^* \\geq C_2 \\geq 11/10 > 30/29$, already at the smallest defect for which such a separation is possible, namely $d = 2$. All parameters are established by the proofs; the only computer-assisted input, the non-list-colourability of a certain 30-vertex, 10-colour list instance $(B,L)$, is certified by an independently checkable DRAT refutation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17557",
  "author_total": 1,
  "authors": [
    {
      "name": "Guillaume Lecomte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6e84zmsktcbx5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17557",
  "fulltext_url": null,
  "id": "6e84zmsktcbx5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17557",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213505105",
  "pdf_url": "https://arxiv.org/pdf/2609.17557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6e84zmsktcbx5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "An improved lower bound for the blowup defective chromatic separation constant",
  "title_text": "An improved lower bound for the blowup defective chromatic separation constant",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:14Z",
  "url": "https://research.draconitic.com/paper/6e84zmsktcbx5",
  "venue": "arXiv",
  "versions": []
}