{
  "abstract": "The directed clique number $\\vecω(T)$ of a tournament $T$ is the minimum, over all orderings of the vertices of $T$, of the clique number of the graph whose edges are the arcs that point backward with respect to the ordering. In this paper, we prove that, for every integer $k \\geq 3$, deciding whether $\\vecω(T) \\leq k$ is NP-complete. This answers a question of Nguyen, Scott, and Seymour, and contrasts with the classical undirected setting, where deciding whether $ω(G) \\leq k$ is polynomial-time solvable for every fixed integer $k$. On the other hand, we give a polynomial-time algorithm distinguishing tournaments with $\\vecω(T) \\leq 2$ from those with $\\vecω(T) > 100$. We also study the tournament analogue of the Gyárfás--Sumner conjecture. We construct new $\\vecχ$-bounding tournaments and thereby prove a conjecture of Aboulker, Aubian, Charbit, and Lopes stating that every class of tournaments with bounded twin-width is $\\vecχ$-bounded. We then exhibit new tournaments that are not $\\vecχ$-bounding, disproving another conjecture of Aboulker, Aubian, Charbit, and Lopes, as well as two conjectures of Kim. Finally, we present infinite families of 3-$\\vecω$-critical and 4-$\\vecω$-critical tournaments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07481",
  "author_total": 2,
  "authors": [
    {
      "name": "Guillaume Aubian"
    },
    {
      "name": "Samuel Coulomb"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0zt3c1y2nvahx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07481",
  "fulltext_url": null,
  "id": "0zt3c1y2nvahx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07481",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212016733",
  "pdf_url": "https://arxiv.org/pdf/2609.07481",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0zt3c1y2nvahx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07481",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07481"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Clique Number of Tournaments II",
  "title_text": "Clique Number of Tournaments II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/0zt3c1y2nvahx",
  "venue": "arXiv",
  "versions": []
}