Meyniel's conjecture on graphs of bounded degree

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 $n^{1\mathrm{/}2-o(1)}$ where $n=\mathrm{\mid }V(G)\mathrm{\mid }$. We also show that proving Meyniel’s conjecture for graphs of bounded degree implies a weak Meyniel’s conjecture for all graphs.