On the cop number of graphs of high girth

TL;DR

This paper derives new lower bounds for the cop number of high girth graphs based on degree and growth conditions, and explores bounds for directed graphs, contributing to understanding pursuit-evasion dynamics.

Contribution

It introduces a novel lower bound for the cop number in high girth graphs and analyzes the bounds' limitations using Ramanujan graphs and expanders.

Findings

Lower bound for cop number: rac{1}{g}( ext{degree}-1)^{ ext{floor}((g-1)/4)}

Bound cannot be improved beyond rac{3}{8}g using Ramanujan graphs

Weak Meyniel's conjecture holds for bounded degree expanders

Abstract

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and minimum degree $\delta$ is at least $\tfrac{1}{g}(\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are also able to prove that it cannot be increased beyond $\frac{3}{8}g$. This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the "weak" Meyniel's conjecture holds for expander graph families of bounded degree.