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.