As it is well known, the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-F\"urer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph $G$ of color multiplicity 4, recognizes whether or not $G$ is identifiable by 2-WL, that is, whether or not 2-WL distinguishes $G$ from any non-isomorphic graph. In fact, we solve the much more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to graphs of color multiplicity 4 with directed and colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-F\"urer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-F\"urer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as $(n_3)$-configurations in incidence geometry.

中文翻译：

---

Weisfeiler-Leman 算法对小颜色类图的可识别性

众所周知，颜色重数最多为 3 的顶点着色图的同构问题可以通过经典的二维 Weisfeiler-Leman 算法 (2-WL) 解决。另一方面，突出的 Cai-F\"urer-Immerman 构造表明，即使是算法的多维版本也不足以处理颜色多重性为 4 的图。我们给出了一个有效的决策程序，给定一个图 $G$ color multiplicity 4, 识别$G$是否可以被2-WL识别，即2-WL是否将$G$与任何非同构图区分开来。实际上，我们解决了更一般的识别问题最大纤维尺寸 4 的给定相干配置是否可分离。这将我们的识别算法扩展到具有有向和有色边缘的颜色多重性 4 的图形。我们的决策程序基于对颜色多重性为 4 的图类的明确描述，这些图不能被 2-WL 识别。颜色多样性 4 的 Cai-F\"urer-Immerman 图在这里明显地作为一个自然子类出现，这表明 Cai-F\"urer-Immerman 构造不是特别的。我们的分类还揭示了对 2-WL 来说很难的其他类型的图。其中之一来自于入射几何中称为 $(n_3)$-configurations 的模式。我们的分类还揭示了对 2-WL 来说很难的其他类型的图。其中之一来自于入射几何中称为 $(n_3)$-configurations 的模式。我们的分类还揭示了对 2-WL 来说很难的其他类型的图。其中之一来自于入射几何中称为 $(n_3)$-configurations 的模式。