An Optimal Lower Bound on the Number of Variables for Graph Identification

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k − 1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices.