{
  "abstract": "An acyclic dicolouring of an oriented graph is a vertex partition in which every colour class and every bipartite subdigraph induced by two classes is acyclic. We first prove the Gallai--Roy-type bound $\\vecχ_a(D)\\leq L(D)$, where $L(D)$ is the maximum order of a directed path. Let $r=\\log_2(8/7)$. Bang-Jensen, Picasarri-Arrieta, and Yeo previously constructed tournaments of order $n$ whose acyclic dichromatic number is at least $n-(\\frac8r)\\log_2 n-\\log_2\\log_2 n$. We improve the leading logarithmic coefficient by a factor of two: for the uniform random tournament $\\mathcal{T}_n$, asymptotically almost surely, $\\vecχ_a(\\mathcal{T}_n)\\geq n-\\frac4r\\log_2 n+\\frac2r\\log_2\\log_2 n-O(1)$. Finally, if $m(k)$ denotes the minimum order of an oriented graph with acyclic dichromatic number at least $k$, tournament completion shows that the same minimum is obtained over tournaments. We prove $m(3)=5$ and $m(4)=7$ and classify the tournament witnesses at these minimum orders: there is one at order five and two at order seven.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22931",
  "author_total": 3,
  "authors": [
    {
      "name": "Yihang Liu"
    },
    {
      "name": "Zhenyu Yang"
    },
    {
      "name": "Yuwan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8sv6c1khewaq5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22931",
  "fulltext_url": null,
  "id": "8sv6c1khewaq5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22931",
  "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": "W7213999018",
  "pdf_url": "https://arxiv.org/pdf/2609.22931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8sv6c1khewaq5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Acyclic Dicolourings of Oriented Graphs: Paths, Random Tournaments, and Critical Orders",
  "title_text": "Acyclic Dicolourings of Oriented Graphs: Paths, Random Tournaments, and Critical Orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8sv6c1khewaq5",
  "venue": "arXiv",
  "versions": []
}