所有至多16个顶点的多面体图都是Schurian的：一项计算验证
All polyhedral graphs with at most 16 vertices are Schurian: a computational verification
AI总结：
该研究通过计算验证了所有至多16个顶点的多面体图（3-连通平面图）均为Schurian，涉及4130亿余个图，结合plantri生成器与1-WL过滤及2-WL比较，为Weisfeiler-Leman维数猜想提供实例证据。
AI中文摘要：
Li、Ponomarenko和Zeman陈述了他们的信念：每个多面体图（即每个3-连通平面图）都是Schurian的，即其相干闭包$WL(X)$与$Aut(X)$的轨道构型一致。结合他们的定理1.1，这将意味着每个多面体图都有至多2的Weisfeiler-Leman维数。我们报告了对所有至多16个顶点的多面体图（共计413,024,847,068个图）的Schurian性质的计算验证，其中没有一个是非Schurian的。该计算结合了生成器plantri与一个单侧1-WL离散性过滤器，并且对于每个通过该过滤器的图，对$\Omega^2$的稳定2-WL划分与$Aut(X)$的轨道划分进行了完整比较。该计算由已发布的锚点计数、两个最大阶段的种子重新枚举、独立的nauty交叉检查以及具有数学固定判决的控制图来保障；代码、检查点和验证脚本作为辅助文件随附本说明。对于$n\le 13$，结果重现了已发布的相干配置枚举所已有的结论，但通过完全不同的途径；从$n=14$起，从该方面推导将需要我们在文献中找不到的步骤，而在$n=16$时，全图陈述完全失败——Shrikhande图是非Schurian的——因此在那里多面体限制承载了结果。该陈述通过穷举计算确立；就一般猜想而言，它是实例证据，而非证明。一篇配套说明处理环面方面，其中在$n=16$时确实出现了非Schurian三角剖分图。
英文摘要：
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 $n = 16$.
