  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.
