{
  "abstract": "We study cyclic orderings of all edges of a complete $k$-uniform hypergraph on $n$ vertices in which the binary incidence sequences of the vertices are cyclic shifts of a common word. The shifts are chosen independently, with no prescribed action on the vertices. For $2\\leq k<n$, coprimality $\\gcd(n,k)=1$ is known to suffice even when consecutive edges must differ by a single vertex exchange. We recall a short orbit construction and prove the converse for the first two nontrivial uniformities without any adjacency requirement. For $k=2$, an ordering exists exactly when $n=2$ or $n$ is odd; for $k=3$, exactly when $n=3$ or $3\\nmid n$. The necessity proofs use reflected convolution identities and pair-intersection counts to constrain the vertex shifts to a torsion coset. For triples, multiplicity-preserving dilation and conditional prime-power capacity bounds complete the argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06724",
  "author_total": 1,
  "authors": [
    {
      "name": "Junyu Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62447d9kk5fbw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06724",
  "fulltext_url": null,
  "id": "62447d9kk5fbw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06724",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212062612",
  "pdf_url": "https://arxiv.org/pdf/2609.06724",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62447d9kk5fbw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06724",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06724"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cyclic Incidence Orderings of Complete Graphs and 3-Uniform Hypergraphs",
  "title_text": "Cyclic Incidence Orderings of Complete Graphs and 3-Uniform Hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/62447d9kk5fbw",
  "venue": "arXiv",
  "versions": []
}