{
  "abstract": "Let $H$ be a simple $3$-uniform hypergraph on $n\\ge7$ vertices, and put $τ_n=\\binom{\\lfloor(n-1)/2\\rfloor}{2}$. We prove that $H$ has a Hamiltonian Berge cycle if its $2$-shadow is $2$-connected, every pair at Berge distance two has an endpoint of hypergraph degree at least $τ_n+1$, and every pair of codegree one has both endpoints above $τ_n$. The distance-two threshold is sharp, and the order range cannot be extended to $n=4,5,6$. The proof separates two structural arguments. A low-degree endpoint of an extremal longest Berge path forces Hamiltonicity, independently of the order and the degree threshold. For high-degree endpoints, injective charging and its equality cases force pairs of codegree one that permit further rotations. The cases $n=7,8$ are settled by exhaustive verification, with a completeness proof and a second implementation using direct vertex--hyperedge cycle search. Keywords: Hamiltonian Berge cycle; Fan-type condition; 3-uniform hypergraph; longest-path rotation; exhaustive verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41577",
  "author_total": 2,
  "authors": [
    {
      "name": "Ailian Chen"
    },
    {
      "name": "Yanqiang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/apze0kh971t1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41577",
  "fulltext_url": null,
  "id": "apze0kh971t1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41577",
  "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/2610.41577",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/apze0kh971t1m.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41577",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41577"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs",
  "title_text": "A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/apze0kh971t1m",
  "venue": "arXiv",
  "versions": []
}