# Cops and robbers on oriented toroidal grids

## TL;DR

This paper investigates the cops and robbers game on specific oriented toroidal grids, proving bounds on the cop number for these structures, which advances understanding of pursuit-evasion dynamics on complex surfaces.

## Contribution

It establishes that the cop number is bounded by a constant for certain oriented toroidal grids, including a specific bound of 13 for k-regularly oriented grids.

## Key findings

- Cop number is bounded by a constant on these graphs.
- Cop number of k-regularly oriented toroidal grids is at most 13.
- Provides new bounds for pursuit-evasion on complex topologies.

## Abstract

The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a constant. We also show that the cop number of every k-regularly oriented toroidal grid is at most 13.