The Weisfeiler-Leman (WL) dimension is an established measure for the inherent descriptive complexity of graphs and relational structures. It corresponds to the number of variables that are needed and sufficient to define the object of interest in a counting version of first-order logic (FO). These bounded-variable counting logics were even candidates to capture graph isomorphism, until a celebrated construction due to Cai, F\"urer, and Immerman [Combinatorica 1992] showed that $\Omega(n)$ variables are required to distinguish all non-isomorphic $n$-vertex graphs. Still, very little is known about the precise number of variables required and sufficient to define every $n$-vertex graph. For the bounded-variable (non-counting) FO fragments, Pikhurko, Veith, and Verbitsky [Discret. Appl. Math. 2006] provided an upper bound of $\frac{n+3}{2}$ and showed that it is essentially tight. Our main result yields that, in the presence of counting quantifiers, $\frac{n}{4} + o(n)$ variables suffice. This shows that counting does allow us to save variables when defining graphs. As an application of our techniques, we also show new bounds in terms of the vertex cover number of the graph. To obtain the results, we introduce a new concept called the WL depth of a graph. We use it to analyze branching trees within the Individualization/Refinement (I/R) paradigm from the domain of isomorphism algorithms. We extend the recursive procedure from the I/R paradigm by the possibility of splitting the graphs into independent parts. Then we bound the depth of the obtained branching trees, which translates into bounds on the WL dimension and thereby on the number of variables that suffice to define the graphs.

翻译：Weisfeiler-Leman（WL）维数是衡量图及关系结构内在描述复杂性的已有标准，它对应于一阶逻辑（FO）计数版本中定义目标对象所需且足够的变量数量。这些有界变量计数逻辑甚至曾被视作捕捉图同构的候选者，直至Cai、Fürer和Immerman [Combinatorica 1992] 提出一项著名构造，表明需要Ω(n)个变量才能区分所有非同构的n顶点图。然而，关于定义每个n顶点图所需且足够的精确变量数量仍知之甚少。对于有界变量（非计数）FO片段，Pikhurko、Veith和Verbitsky [Discret. Appl. Math. 2006] 给出了(n+3)/2的上界，并证明该界本质上是紧的。我们的主要结果表明，在计数量词存在的情况下，(n/4)+o(n)个变量便足够。这说明计数机制在定义图时确实能够节省变量数量。作为这些技术的应用，我们还基于图的顶点覆盖数给出了新界。为获得这些结果，我们引入了称为图WL深度这一新概念，用以分析同构算法领域中个性化/精化（I/R）范式内的分支树。我们将I/R范式中的递归过程扩展为将图拆分为独立部分的可能，进而界定了所得分支树的深度，这转化为对WL维数的界，从而也转化为定义图所需足够变量数量的界。