k-planar crossing number of random graphs and random regular graphs

We give an explicit extension of Spencer's result on the biplanar crossing number of the Erdős-Rényi random graph G(n,p). In particular, we show that the k-planar crossing number of G(n,p) is almost surely Ω((n2p)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 Ω((dn)2) for d>c0 for some constant c0=c0(k).