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

TL;DR

This paper extends known results on the crossing number of random graphs, showing that the k-planar crossing number is almost surely large for Erdős-Rényi and random regular graphs, indicating high complexity in their planar representations.

Contribution

It provides explicit bounds on the k-planar crossing number for both Erdős-Rényi and random regular graphs, extending previous results to these models.

Findings

k-planar crossing number of G(n,p) is almost surely Ω((n^2 p)^2)

k-planar crossing number of random d-regular graphs is Ω((dn)^2) for large d

Results hold for fixed k and large n

Abstract

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 $c_0=c_0(k)$.