Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Let v(F ) denote the number of vertices in a fixed connected pattern graph F . We show an infinite family of patterns F such that the existence of a subgraph isomorphic to F is expressible by a first-order sentence of quantifier depth 23 v(F ) + 1, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any F with using less than 2 3 v(F ) − 2 first-order variables.