通过游走计数对随机循环图进行规范化
摘要
众所周知,几乎所有图都可通过称为颜色细化(亦称一维 Weisfeiler-Leman 算法)的简单组合例程规范化。该方法以高概率给随机输入图的每个顶点分配唯一标签,因此仅适用于非对称图。若输入图高度对称,组合细化技术的效力便成为一个微妙问题。我们证明颜色细化与顶点个体化相结合,为几乎所有循环有向图(即循环群的 Cayley 有向图)给出了规范标记。该结果首次提供了组合细化在顶点传递图类中良好平均case性能的证据。值得注意的是,我们甚至不需要颜色细化算法的全部能力。我们展示顶点 的规范标签仅通过计数从 到某个体化顶点的各长度游走即可获得。我们的分析还意味着几乎所有循环图在 Tinhofer 意义下是紧的,即其分数自同构多胞形是整的。最后,我们展示对几乎所有循环图,可通过更强的二维 Weisfeiler-Leman 算法构造规范的 Cayley 表示。
引用
@article{arxiv.2310.05788,
title = {Canonization of a random circulant graph by counting walks},
author = {Oleg Verbitsky and Maksim Zhukovskii},
journal= {arXiv preprint arXiv:2310.05788},
year = {2025}
}
备注
35 pages. A preliminary version of this paper appeared in the Proceedings of the 18th International Conference and Workshops on Algorithms and Computation (WALCOM'24), published in Lecture Notes in Computer Science Vol. 14549, Springer 2024. Theorem 1.2 as well as Corollaries 2.4 and 2.5 are new. Section 5 is extended by including more technical and expository details