{
  "abstract": "For each integer $k\\ge2$, let $C(k,k)$ denote the oriented cycle formed by two internally vertex-disjoint directed paths of length $k$ with the same initial and terminal vertices. Picasarri-Arrieta and Rambaud ({\\small European J. Combin., 2024}) asked for the smallest integer $g_k$ such that every digraph with minimum out-degree at least two and digirth at least $g_k$ contains a subdivision of $C(k,k)$. We completely resolve this problem by proving that $g_k=k+1$. Our proof relies on a refinement of Thomassen's theorem ({\\small Discrete Math., 1985}) on two directed cycles intersecting in exactly one vertex: we prove that every digraph with minimum out-degree at least two contains two directed cycles intersecting in exactly one arc. We also establish an embedding theorem for out-forests with prescribed images of their roots, under minimum out-degree and digirth conditions determined by the number of leaves and the heights of vertices in antichains. As an application, for integers $k,\\ell\\ge2$, we show that every digraph with minimum out-degree at least \\(k\\) and digirth at least \\(k(\\ell-1)+1\\) contains \\(k\\) directed paths of length \\(\\ell\\) sharing only their initial vertex, which can be prescribed arbitrarily. This improves the previous digirth bound \\((k^\\ell-1)/(k-1)+1\\) of Picasarri-Arrieta and Rambaud to a sharp bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38574",
  "author_total": 6,
  "authors": [
    {
      "name": "Bin Chen"
    },
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Xinmin Hou"
    },
    {
      "name": "Siyue Liu"
    },
    {
      "name": "Xinjie Wang"
    },
    {
      "name": "Xinyu Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z9wmvaj5dd71.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38574",
  "fulltext_url": null,
  "id": "7z9wmvaj5dd71",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38574",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z9wmvaj5dd71.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The exact digirth and out-degree bounds for $C(k,k)$-subdivision and out-forest in digraphs",
  "title_text": "The exact digirth and out-degree bounds for C(k,k)-subdivision and out-forest in digraphs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7z9wmvaj5dd71",
  "venue": "arXiv",
  "versions": []
}