# On the cop number of toroidal graphs

## TL;DR

This paper proves that the maximum number of cops needed to catch a robber on any toroidal graph is three, resolving a long-standing conjecture in graph theory.

## Contribution

It establishes the first proof that the cop number for all toroidal graphs is at most three, confirming a conjecture from 2001.

## Key findings

- Cop number of toroidal graphs is at most 3
- Resolved a conjecture by Schroeder from 2001
- Addresses a question posed by Andreae in 1986

## Abstract

We show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which is implicit in a question by Andreae from 1986.