  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.
