Meyniel Extremal Families of Abelian Cayley Graphs

TL;DR

This paper constructs specific abelian Cayley graphs demonstrating that the known upper bound on the cop number in the Cops and Robbers game is tight, providing evidence supporting Meyniel's conjecture.

Contribution

It introduces families of abelian Cayley graphs that are Meyniel extremal, showing the upper bound on cop number is tight for these graphs.

Findings

Constructed Cayley graphs with   generators that are K_{2,3}-free.

Proved the tightness of the  bound for Cayley graphs.

Established the tightness of the Kf6ve1ri-Sf3s-Ture1n theorem for 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 $O(\sqrt{n})$, i.e., that $O(\sqrt{n})$ 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 is tight up to a multiplicative constant. In particular, this shows that Meyniel's conjecture, if true, is tight to a multiplicative constant 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 K\"{o}v\'{a}ri, S\'{o}s, and Tur\'{a}n 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.