Meyniel's conjecture on graphs of bounded degree

TL;DR

This paper investigates Meyniel's conjecture within the context of graphs with bounded degree, establishing new lower bounds on cop numbers for subcubic graphs and linking bounded degree cases to the general conjecture.

Contribution

It proves that subcubic graphs can have cop numbers growing at least as fast as the square root of their size and connects Meyniel's conjecture for bounded degree graphs to the general case.

Findings

Subcubic graphs can have cop number at least n^{1/2-o(1)}.

Proving Meyniel's conjecture for bounded degree graphs implies a weaker form for all graphs.

Established a lower bound on cop numbers in graphs of bounded degree.

Abstract

The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set $\{c(G) \mid G \text{ cubic}\}$ is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1/2-o(1)}$ where $n=|V(G)|$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.