K-planar Crossing Number of Random Graphs and Random Regular Graphs

@article{Asplund2018KplanarCN,
  title={K-planar Crossing Number of Random Graphs and Random Regular Graphs},
  author={J. Asplund and Thao T. Do and Arran Hamm and L. Sz{\'e}kely and Libby Taylor and Zhiyu Wang},
  journal={Discret. Appl. Math.},
  year={2018},
  volume={247},
  pages={419-422}
}
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 $c_0=c_0(k)$. 
Using Block Designs in Crossing Number Bounds.
The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$,Expand

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