# 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 $c_0=c_0(k)$.

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Using Block Designs in Crossing Number Bounds.

- Mathematics
- 2018

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

#### References

SHOWING 1-6 OF 6 REFERENCES

Applications of the crossing number

- Mathematics, Computer Science
- Algorithmica
- 2005

A partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others, where any piecewise linear one-to-one mappingf∶R2→R2 withf(pi)=qi (1≤i≤n) is composed of at least Ω(n2) linear pieces. Expand

A proof of Alon's second eigenvalue conjecture and related problems

- Mathematics, Computer Science
- ArXiv
- 2004

These theorems resolve the conjecture of Alon, that says that for any > 0a ndd, the second largest eigenvalue of \ most" random dregular graphs are at most 2 p d 1+ (Alon did not specify precisely what \most" should mean or what model of random graph one should take). Expand

Explicit construction of linear sized tolerant networks

- Computer Science, Mathematics
- Discret. Math.
- 1988

This paper constructs explicitly graphs with O(m/@e) vertices and maximum degree O(1/@ e^2), such that after removing any (1-@e%) portion of their vertices or edges, the remaining graph still contains a path of length m. Expand

The Probabilistic Method

- Computer Science
- SODA
- 1992

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored. Expand

Fault diagnosis in a small constant number of parallel testing rounds

- Computer Science
- SPAA '93
- 1993

A parallel algorithm is provided that determines which processors are good and which are faulty in 32 rounds of testing, pre Tided that a strict majority of the processor are good. Expand