  We give an explicit extension of Spencer's result on the biplanar crossing
number of the Erdos-Renyi random graph $G(n,p)$. In particular, we show that
the k-planar crossing number of $G(n,p)$ is almost surely $\Omega((n^2p)^2)$.
Along the same lines, we prove that for any fixed $k$, the $k$-planar crossing
number of various models of random $d$-regular graphs is $\Omega ((dn)^2)$ for
$d > c_0$ for some constant $c_0=c_0(k)$.
