{
  "schema_version": 1,
  "problem_number": "OWR-11786-023",
  "title": "A Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let f̄(2,n) be the maximum diameter of the dual graph of a triangulated surface with boundary on n vertices. At a 2012 Oberwolfach workshop, Santos asked whether f̄(2,n) ≤ n − 3. We show that the answer is no. A triangulation M₇ of the real projective plane minus two disjoint open discs has 7 vertices, 10 triangles and dual diameter 5. The same complex appears in work of Holmes on Stanley–Reisner rings with Serre's property (S₂), where it is not identified as a surface. Gluing copies of M₇ along boundary edges gives f̄(2,n) ≥ n − 3 + ⌊(n − 2)/5⌋ for all n ≥ 3, so the excess over n − 3 is unbounded. An orientable surface of genus 2 with 10 vertices and dual diameter 8 gives the bound n − 3 + ⌊(n − 2)/8⌋ for orientable surfaces. A short layering argument shows f̄(2,n) ≤ max(n − 3, 2n − 8), and a theorem of Holmes gives max(2n − 10, n − 2). Computer searches with DRAT-certified unsatisfiability proofs, confirmed by an exhaustive enumeration of the surfaces with at most 10 vertices, show that f̄(2,n) = n − 3 for n ≤ 6 and f̄(2,n) = n − 2 for 7 ≤ n ≤ 10 (for 6 ≤ n ≤ 9 these values also follow from results of Holmes), that M₇ is the only counterexample with at most 7 vertices, and that orientable surfaces with at most 9 vertices satisfy the bound. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "keywords": [
    "triangulated surface",
    "surface with boundary",
    "dual graph",
    "dual diameter",
    "Hirsch conjecture",
    "Hirsch bound",
    "counterexample",
    "normal complexes",
    "Stanley–Reisner rings",
    "SAT solving",
    "DRAT proofs",
    "exhaustive enumeration",
    "Oberwolfach Reports",
    "OWR-11786-023",
    "math.CO",
    "math.GT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-01",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-11786-023/",
  "pdf_url": "https://eulersolve.org/papers/owr-11786-023/paper.pdf?v=af2552ed3b11",
  "doi": "10.5281/zenodo.23072344",
  "zenodo_record_url": "https://zenodo.org/records/23072344",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Santos' question (OWR 24/2012, p. 1480) negatively: a 7-vertex surface with boundary has dual diameter 5 > n − 3, and the excess over n − 3 is unbounded. The 7-vertex complex appears in work of Holmes (EJC 2018), whose results already give the small values for 6 ≤ n ≤ 9; the note identifies the complexes as surfaces and adds the unbounded excess, orientable examples and certified values up to n = 10. The asymptotic slope and the orientable growth remain open.",
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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."
}