In this paper, we show that the $(3k+4)$-dimensional Weisfeiler--Leman algorithm can identify graphs of treewidth $k$ in $O(\log n)$ rounds. This improves the result of Grohe & Verbitsky (ICALP 2006), who previously established the analogous result for $(4k+3)$-dimensional Weisfeiler--Leman. In light of the equivalence between Weisfeiler--Leman and the logic $\textsf{FO} + \textsf{C}$ (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an improvement in the descriptive complexity for graphs of treewidth $k$. Precisely, if $G$ is a graph of treewidth $k$, then there exists a $(3k+5)$-variable formula $\varphi$ in $\textsf{FO} + \textsf{C}$ with quantifier depth $O(\log n)$ that identifies $G$ up to isomorphism.

翻译：本文证明，$(3k+4)$-维魏斯费勒-莱曼算法可以在$O(\log n)$轮内识别树宽为$k$的图。这一结果改进了Grohe与Verbitsky（ICALP 2006）先前对$(4k+3)$-维魏斯费勒-莱曼算法的类似结论。基于魏斯费勒-莱曼算法与逻辑语言$\textsf{FO} + \textsf{C}$（Cai、F\"urer与Immerman，Combinatorica 1992）的等价性，我们改进了树宽为$k$的图的描述复杂性。具体而言，若$G$是树宽为$k$的图，则存在一个量词深度为$O(\log n)$的$(3k+5)$-变量公式$\varphi$（属于$\textsf{FO} + \textsf{C}$），可在同构意义下唯一标识$G$。