It is well known that almost all graphs are canonizable by a simple combinatorial routine known as color refinement. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of color refinement with vertex individualization produces a canonical labeling for almost all circulant digraphs (Cayley digraphs of a cyclic group). To our best knowledge, this is the first application of combinatorial refinement in the realm of vertex-transitive graphs. Remarkably, we do not even need the full power of the color refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are canonizable by Tinhofer's canonization procedure. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the 2-dimensional Weisfeiler-Leman algorithm.

翻译：众所周知，几乎所有的图都可以通过一种称为颜色细化的简单组合方法进行正则化。该方法以高概率为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。若输入图具有高度对称性，组合细化技术的有效性则成为一个微妙问题。我们证明，颜色细化与顶点个体化相结合，可为几乎所有的循环有向图（循环群的凯莱有向图）产生正则标号。据我们所知，这是组合细化技术在顶点传递图领域的首次应用。值得注意的是，我们甚至不需要颜色细化算法的全部能力。研究表明，顶点$v$的正则标号仅需通过计算从$v$到个体化顶点的各长度游走数量即可获得。我们的分析还表明，几乎所有的循环图都可以通过Tinhofer的正则化过程实现正则化。最后，我们证明对于几乎所有的循环图，可以通过二维Weisfeiler-Leman算法构建正则凯莱表示。