{
  "schema_version": 1,
  "problem_number": "OWR-14299911-020",
  "title": "The Travelling Polytope Constant of a Product of Trees: An Answer to a Question of Genevois",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The travelling polytope constant TP(X) of a graph X compares the travelling salesman problem through a finite set F of vertices with the shortest way to sweep F by convex hulls which grow or shrink by one adjacent vertex at a time. A. Genevois introduced it in his thesis (2017), as the travelling salesman constant of a CAT(0) cube complex; he proved that it equals d/(2d − 1) for ℤ^d and lies in [1/2, 2/3] ∪ {1} for unbounded median graphs of bounded degree, and asked for its value for a product of two trees. At the Oberwolfach workshop Median Geometry and Applications (2026) he asked again to compute it for products of (regular) trees. We show that TP(T₁ × ⋯ × T_d) = d′/(2d′ − 1), where d′ ≥ 1 is the number of factors of unbounded diameter, without any assumption on the degrees, and treat the remaining cases, in which all factors have finite diameter. So a product of d regular trees of degree at least 2 has constant d/(2d − 1), as Genevois's guess for cocompact cube complexes predicts. The upper bound is Genevois's grid example for ℤ^d, placed on geodesic segments of the factors. The new part is the lower bound: by Genevois's formula, the shortest sweep has twice as many steps as there are hyperplanes separating two of the points, and Euler tours of the projected hulls give a 1-Lipschitz map from a box onto the hull of the points, through which the classical strip bound for the travelling salesman problem in a cube applies. The values obtained were known values of the constant, so the note gives no new value; which values the constant takes on median graphs remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GR",
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "travelling polytope constant",
    "travelling salesman constant",
    "median graphs",
    "CAT(0) cube complexes",
    "products of trees",
    "regular trees",
    "convex hulls",
    "hyperplanes",
    "Euler tour",
    "strip method",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299911-020",
    "math.GR",
    "math.MG",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299911-020/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299911-020/paper.pdf?v=8fb47f088314",
  "doi": "10.5281/zenodo.23430048",
  "zenodo_record_url": "https://zenodo.org/records/23430048",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Computes the constant for all products of trees (Genevois's thesis Question 12.17); no new value of the constant; which values it takes on median graphs remains 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."
}