Most Generalized Petersen graphs of girth 8 have cop number 4

TL;DR

This paper proves that most generalized Petersen graphs with girth 8 have a cop number of 4, extending previous bounds and providing new conditions for cop number in cubic graphs of high girth.

Contribution

It establishes that, except for specific cases, generalized Petersen graphs of girth 8 have cop number 4, and generalizes to graphs with girth at least 8 or 9.

Findings

Most girth 8 generalized Petersen graphs have cop number 4.

Graphs with girth ≥9 and minimum degree δ have cop number at least δ+1.

The result applies to a broad class of cubic graphs with high girth.

Abstract

A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$ vertices (the parameter $k$ is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of $GP(n,k)$ is at most $4$, for all permissible values of $n$ and $k$. In this paper we prove that the cop number of "most" generalized Petersen graphs is exactly $4$. More precisely, we show that unless $n$ and $k$ fall into certain specified categories, then the cop number of $GP(n,k)$ is $4$. The graphs to which our result applies all have girth $8$. In fact, our argument is slightly more general: we show that in any cubic graph of girth at least $8$, unless there exist two cycles of length $8$ whose intersection is a path of length $2$, then the cop number of the graph is at least $4$. Even more generally, in a graph of girth at least $9$ and minimum valency $\delta$, the cop number is at least $\delta+1$.