Cops and robbers on directed and undirected abelian Cayley graphs

TL;DR

This paper establishes new upper bounds on the cop number for directed and undirected Cayley graphs on abelian groups, using a refined inductive method, and constructs extremal families with cop number proportional to the square root of the number of vertices.

Contribution

It introduces a refined inductive approach to bound the cop number and constructs new Meyniel extremal families within Cayley graphs on abelian groups.

Findings

Upper bound of O(√n) for cop number on Cayley graphs on abelian groups

Improved previous bounds for undirected Cayley graphs

Construction of Meyniel extremal families with cop number Θ(√n)

Abstract

We show that the cop number of directed and undirected Cayley graphs on abelian groups has an upper bound of the form of $O(\sqrt{n})$, where $n$ is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new \emph{Meyniel extremal families}, which contain graphs of every order $n$ with cop number $\Theta(\sqrt{n})$.