An Upper Bound on the Weisfeiler-Leman Dimension

Abstract: The Weisfeiler-Leman (WL) dimension is a standard measure in descriptive complexity theory for the structural complexity of a graph. We prove that the WL-dimension of a graph on $n$ vertices is at most $3/20 \cdot n + o(n)= 0.15 \cdot n + o(n)$. The proof develops various techniques to analyze the structure of coherent configurations. This includes sufficient conditions under which a fiber can be restored up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an analysis of interspaces involving small fibers. As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size.