All polyhedral graphs with at most 16 vertices are Schurian: a computational verification

@misc{reh2026polyhedral,
  title = {{All polyhedral graphs with at most 16 vertices are Schurian: a computational verification}},
  author = {David Reh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03729},
  eprint = {2610.03729},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03729},
}

TY  - UNPB
TI  - All polyhedral graphs with at most 16 vertices are Schurian: a computational verification
AU  - David Reh
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.03729
UR  - https://arxiv.org/abs/2610.03729
AB  - 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≤ 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.
ER  -