{
  "schema_version": 1,
  "problem_number": "OPG-37325",
  "title": "Logarithmic Equivalence Covers of Powers of Cycles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An equivalence graph is a vertex-disjoint union of cliques. We give an explicit cover of every noncomplete cycle power by a logarithmic number of equivalence subgraphs. More precisely, for integers k>=1 and n>=2k+2, ceil(log2(2k+2)) <= eq(C_n^k) <= 4 ceil(log2(k+1))+1. The upper bound partitions the cycle into short clique blocks and uses binary encodings for threshold adjacency between blocks. The lower bound is an application of Alon's multilinear rank method. Consequently the equivalence covering number has order log(k+1) uniformly in n, giving a negative answer to the linear-growth conjecture recorded in Open Problem Garden, including its intended large-n regime. We credit earlier logarithmic co-chain encodings and record a related 2010 conference abstract whose numerical bounds were not available in the located text. No absolute priority claim is made.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "equivalence covering",
    "cycle power",
    "co-chain graph",
    "binary encoding",
    "graph covering",
    "rank bound",
    "OPG-37325",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-37325/",
  "pdf_url": "https://eulersolve.org/papers/opg-37325/paper.pdf?v=edc4fe4f7b32",
  "doi": "10.5281/zenodo.23042422",
  "zenodo_record_url": "https://zenodo.org/records/23042422",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Not exact covering numbers or an optimal leading constant. Alon rank method and GMR co-chain encoding credited; numerical overlap with a 2010 MathFest abstract unresolved, so no absolute priority claim. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "edc4fe4f7b3226160005f5060a55f6e598622af31db77648ac28d299c9fd163b"
    },
    "source.zip": {
      "sha256": "893f1f6ec417aa774deab813622df2ad4c95a9b444614538cf90f9c80f03e6d9"
    },
    "verification_report.md": {
      "sha256": "895553b0c41605901cfd2605d61c8f35c97c5ca23f57a8c75baa8eb6686b6c99"
    }
  },
  "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."
}