{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0177",
  "title": "A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give two loopless directed graph layers on six vertices, each with constant outdegree two, having no rainbow directed cycle and hence no increasing rainbow cycle. For the fixed label order 1<2, the sizes of the sets reachable by possibly trivial increasing paths are 5,5,5,5,5,4. Their average is 29/6<5=1+delta_1+delta_2. Thus the uniform-start-vertex interpretation of the ordered-reachability conjecture attributed to DeVos in Sullivan's 2006 survey is false. Uniform blow-ups give examples on 6m vertices with average 1+23m/6, below the proposed bound 1+4m for every m>=1. The proof is an explicit neighborhood calculation and does not depend on the numerical search that located the example.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "ordered reachability",
    "rainbow cycles",
    "directed graphs",
    "counterexample",
    "AIM-COMBINATORICS-0177",
    "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/aim-combinatorics-0177/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0177/paper.pdf?v=ddd2cd0c85f6",
  "doi": "10.5281/zenodo.23016087",
  "zenodo_record_url": "https://zenodo.org/records/23016087",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The original source does not specify its averaging variable. This counterexample addresses a fixed label order and a uniform average over starting vertices, not averaging over label orders: the reverse order gives reachability total 35 rather than 29. It does not disprove the Caccetta-Haggkvist conjecture or its unordered rainbow-path analogue, and no minimal-size claim is made. Self-audited, unrefereed; no formal verification or absolute priority certificate is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ddd2cd0c85f639ee5cfbcab852fbee33e1d6232237b6ce2468498de77156516c"
    },
    "source.zip": {
      "sha256": "eaa8818cc2e59284b965c4f25057618486053b43cf1611e0e3107aa87a0fee60"
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    "verification_report.md": {
      "sha256": "31a51fcd1b84f68462fccef560c3d82c2c0b89475ca2c6a81cb7f0c4409f49a0"
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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."
}