{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0181",
  "title": "Cyclic Counterexamples to the Exact-Distance Higher-Neighborhood Conjecture",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every integer k>=3 and n>k(k+1), the oriented cyclic Cayley graph with increments 1,k,k+1 has directed girth greater than k, yet every vertex has 2k-1 vertices at shortest directed distance k-1 and only k+1 at distance k. This gives complete counterexamples to the exact-distance higher-neighborhood assertion printed as Conjecture 6.13 in Sullivan's 2006 survey, corresponding to AIM-COMBINATORICS-0181. The graphs are strongly connected, vertex-transitive, and three-in/three-out regular, with arbitrarily large directed girth at each fixed k. Independent blow-ups yield unbounded degrees and deficits. An elementary integer sumset identity proves the general construction; a thirteen-vertex example and 3,546 exact checks per execution mode supplement the proof. The k=2 second-neighborhood problem and a different recursive external-neighborhood formulation are not resolved. AI-assisted, self-audited and unrefereed; no formal verification, independent peer review, global minimality, or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "directed graph",
    "exact-distance neighborhood",
    "Cayley graph",
    "directed girth",
    "Burckel conjecture",
    "AIM-COMBINATORICS-0181"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0181/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0181/paper.pdf?v=9d34449e0673",
  "doi": "10.5281/zenodo.23445795",
  "zenodo_record_url": "https://zenodo.org/records/23445795",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete refutation of Sullivan Conjecture 6.13 with exact shortest-distance neighborhoods, for every k>=3. The k=2 Seymour specialization is not resolved, and no refutation of the distinct recursive external-neighborhood formulation is claimed. Thirteen is the displayed witness order, not a certified global minimum. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
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      "sha256": "9d34449e0673c8ca6c954adbfaa033e9679d73f837964928bee2c6ae45086c36"
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    "source.zip": {
      "sha256": "39e6369b084533c17af10bf27e7712231de99c35c3d674b2f2693dfda6a51925"
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    "verification_report.md": {
      "sha256": "f160048d494d1604ffcc314905887171f4ab46c5e8983e3bfc5bd820869bcb9b"
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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."
}