{
  "schema_version": 1,
  "problem_number": "OPG-60013",
  "title": "Weak Saturation of the Cube in the Clique: Exact Values for n ≤ 13 and Linear Bounds",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For graphs G and H, wsat(G, H) is the least number of edges of a spanning subgraph F of G such that the edges of G missing from F can be added one at a time, each completing a new copy of H. Morrison and Noel asked (Open Problem Garden, 2016) to determine wsat(K_n, Q_3), where Q_3 is the cube. We give a partial answer. For n ≤ 7 no edge can be added and the value is n(n − 1)/2. By exhaustive computer searches, each carried out with at least two independent programs, wsat(K_n, Q_3) = 15, 16, 18, 19, 21, 23 for n = 8, 9, 10, 11, 12, 13; extremal graphs are given explicitly, with certificates that can be checked without a computer. The values for n ≤ 11 and the upper bound wsat(K_n, Q_3) ≤ ⌊(7n + 2)/4⌋ for n ≥ 9, which comes from a gadget with four vertices and seven edges, were obtained earlier in two unrefereed preprints (Carptopus, 2026; V. Saha and N. Saha, 2026); the values for n = 12, 13 are new. We prove wsat(K_n, Q_3) ≥ (5n − 7)/3 for n ≥ 8 by the cluster method of Terekhov and Zhukovskii, with clusters that may share a vertex. Hence the limit of wsat(K_n, Q_3)/n, which exists by a result of Alon, lies in [5/3, 7/4]; the lower end improves the value 11/7 given by a general theorem of Terekhov and Zhukovskii. We conjecture, on computational evidence only, that wsat(K_n, Q_3) = ⌊(7n + 2)/4⌋ for all n ≥ 9. The exact value for n ≥ 14 remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "weak saturation",
    "weakly saturated graphs",
    "graph bootstrap percolation",
    "hypercube",
    "cube Q3",
    "complete graph",
    "computer-assisted proof",
    "exhaustive search",
    "partial answer",
    "Open Problem Garden",
    "UnsolvedMath",
    "OPG-60013",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-60013/",
  "pdf_url": "https://eulersolve.org/papers/opg-60013/paper.pdf?v=8fcd05491024",
  "doi": "10.5281/zenodo.23483625",
  "zenodo_record_url": "https://zenodo.org/records/23483625",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer: the values for n <= 11 and the upper bound appeared earlier in two unrefereed preprints (Carptopus 2026; V. Saha and N. Saha 2026); new here are n = 12, 13 and the lower bound (5n-7)/3. The exact value for n >= 14 is open.",
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    "source.zip": {
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    "verification_report.md": {
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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."
}