Meyniel Extremal Families of Abelian Cayley Graphs

Abstract

We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel’s conjecture states that for every connected graph G on n vertices, the cop number of G is upper bounded by $$O(\sqrt{n})$$ . That is, for every graph G on n vertices $$O(\sqrt{n})$$ cops suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is $$O(\sqrt{n})$$ . This proves that the $$O(\sqrt{n})$$ upper bound for Cayley graphs proved by Bradshaw (Discret Math 343:1, 2019) is tight. In particular, this shows that Meyniel’s conjecture, if true, is tight even for abelian Cayley graphs. In order to prove the result, we construct Cayley graphs on n vertices with $$\Omega (\sqrt{n})$$ generators that are $$K_{2,3}$$ -free. This shows that the Kövári, Sós, and Turán theorem, stating that any $$K_{2,3}$$ -free graph of n vertices has at most $$O(n^{3/2})$$ edges, is tight up to a multiplicative constant even for abelian Cayley graphs.