  In the game of cops and robber, the cops try to capture a robber moving on
the vertices of the graph. The minimum number of cops required to win on a
given graph $G$ is called the cop number of $G$. The biggest open conjecture in
this area is the one of Meyniel, which asserts that for some absolute constant
$C$, the cop number of every connected graph $G$ is at most $C \sqrt{|V(G)|}$.
In a separate paper, we showed that Meyniel's conjecture holds asymptotically
almost surely for the binomial random graph. The result was obtained by showing
that the conjecture holds for a general class of graphs with some specific
expansion-type properties. In this paper, this deterministic result is used to
show that the conjecture holds asymptotically almost surely for random
$d$-regular graphs when $d = d(n) \ge 3$.
