{
  "abstract": "For a tournament $T$, let $ω(T)$ be the minimum clique number among the backedge graphs of $T$, and let $χ(T)$ be its dichromatic number. We give a template-lifting construction. It turns a $k$-template into a regular, vertex-transitive Cayley tournament that is simultaneously $(k+1)$-$ω$-critical and $(k+1)$-$χ$-critical. The output is also a $(k+1)$-template. Iterating the construction, we prove that for every $k\\geq3$, there is a positive even integer $m_k$ with the following property. Every $N>1$ with $N\\equiv1\\pmod{m_k}$ is the order of a regular, vertex-transitive Cayley tournament that is simultaneously $k$-$ω$-critical and $k$-$χ$-critical. This proves a conjecture of Aboulker, Aubian, Charbit, and Lopes and gives a negative answer to their bounded-certificate question when the hypothesis is $ω(T)\\geq k$. We also find the clique number of a cyclic substitution when each block satisfies $ω=χ$. We then describe exactly when this substitution is $ω$-critical if the blocks are $χ$-critical and satisfy $ω=χ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08658",
  "author_total": 2,
  "authors": [
    {
      "name": "Guantao Chen"
    },
    {
      "name": "Shengze Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/emzqpk6a10ysk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08658",
  "fulltext_url": null,
  "id": "emzqpk6a10ysk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08658",
  "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": "W7211995460",
  "pdf_url": "https://arxiv.org/pdf/2609.08658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/emzqpk6a10ysk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers",
  "title_text": "Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/emzqpk6a10ysk",
  "venue": "arXiv",
  "versions": []
}