{
  "schema_version": 1,
  "problem_number": "OWR-1782-009",
  "title": "Small Counterexamples to the All-n Form of the Exact Codegree Conjecture for Tight Hamiltonian Cycles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Rödl, Ruciński and Szemerédi stated the following conjecture, which they attribute to Katona and Kierstead: every k-uniform hypergraph on n ≥ k+1 ≥ 4 vertices in which every (k−1)-set lies in at least ⌊(n−k+3)/2⌋ edges has a tight Hamiltonian cycle. They proved it for k = 3 and all sufficiently large n. We observe that the statement, as written for all n ≥ k+1, fails for small n. The smallest counterexample is a 3-graph on 7 vertices: an apex joined to all pairs of a 6-set, together with the ten faces of the hemi-icosahedron on that set. It has minimum codegree 3 = ⌊7/2⌋ but no tight Hamiltonian cycle, and the proof is two lines. A variant of the construction gives counterexamples for k = 4 and k = 5 on k+4 vertices. A computer search finds two more, on 9 vertices for k = 3 and on 10 vertices for k = 5; the second meets the bound (n−k+3)/2 even without the floor. These are exceptions for small n only, and the asymptotic statement is not affected. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE_ALL_N_FORM",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "hypergraphs",
    "tight Hamiltonian cycles",
    "minimum codegree",
    "Dirac-type theorems",
    "hemi-icosahedron",
    "counterexamples",
    "Oberwolfach Reports",
    "OWR-1782-009",
    "OWR-1386-013",
    "open mathematics",
    "mathematical proof",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1782-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-1782-009/paper.pdf?v=ee3b8aa85751",
  "doi": "10.5281/zenodo.23006718",
  "zenodo_record_url": "https://zenodo.org/records/23006718",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The note refutes the conjecture only in its all-n form. The large-n statement, which is the substance of the conjecture, is not affected: it was proved for k = 3 by Rödl, Ruciński and Szemerédi and has been announced for all k. The note does not claim to disprove the Katona–Kierstead conjecture.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}