On Generalised Petersen Graphs of Girth 7 that have Cop Number 4

TL;DR

This paper characterizes when generalized Petersen graphs with girth 7 have a cop number of 4, expanding understanding of their structural properties and confirming previous bounds for specific cases.

Contribution

It identifies exact conditions on n and k for girth 7 Petersen graphs to have cop number 4, explaining previously unclassified cases and situating them within infinite families.

Findings

Cop number is 4 when n=7k/i with i in {1,2,3}.

Provides a complete classification for girth 7 generalized Petersen graphs with cop number 4.

Extends previous results that only covered girth 8 cases.

Abstract

We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most $4$. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number $4$ but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number $4$ and girth $8$, while this paper explains those that have girth $7$.)