{
  "abstract": "Li, Ponomarenko and Zeman state their belief that every polyhedral graph (i.e. every $3$-connected planar graph) is Schurian: that its coherent closure $WL(X)$ coincides with the orbital configuration of $Aut(X)$. Combined with their Theorem 1.1, this would imply that every polyhedral graph has Weisfeiler-Leman dimension at most $2$. We report a computational verification of the Schurian property for all polyhedral graphs with at most $16$ vertices, a total of 413,024,847,068 graphs, none of which is non-Schurian. The computation combines the generator plantri with a one-sided $1$-WL discreteness filter and, for every graph surviving that filter, a full comparison of the stable $2$-WL partition of $Ω^2$ with the orbit partition of $Aut(X)$. The computation is guarded by the published anchor counts, a seeded re-enumeration of the two largest stages, independent nauty cross-checks and control graphs with mathematically fixed verdicts; code, checkpoints and a verification script accompany the note as ancillary files. For $n\\le 13$ the result reproduces what already follows from the published enumeration of coherent configurations, by an entirely different route; from $n=14$ on, deriving it from that side would require a step we could not find in the literature, and at $n=16$ the all-graphs statement fails outright - the Shrikhande graph is non-Schurian - so there the polyhedral restriction carries the result. The statement is established by exhaustive computation; with respect to the general conjecture it is instance evidence, not a proof. A companion note treats the torus side, where non-Schurian triangulation graphs do occur at $n = 16$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03729",
  "author_total": 1,
  "authors": [
    {
      "name": "David Reh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b0ebpf18qaxhj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03729",
  "fulltext_url": null,
  "id": "b0ebpf18qaxhj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03729",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7220709515",
  "pdf_url": "https://arxiv.org/pdf/2610.03729",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b0ebpf18qaxhj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03729",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03729"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "All polyhedral graphs with at most 16 vertices are Schurian: a computational verification",
  "title_text": "All polyhedral graphs with at most 16 vertices are Schurian: a computational verification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b0ebpf18qaxhj",
  "venue": "arXiv",
  "versions": []
}