Note on k-planar crossing numbers

TL;DR

This paper investigates the k-planar crossing number of graphs, establishing bounds on how the crossing number can be partitioned across multiple subgraphs, with implications for rectilinear variants.

Contribution

It provides a new upper bound for the k-planar crossing number relative to the original crossing number, and shows the bound's tightness with respect to smaller constants.

Findings

For every k ≥ 1, cr_k(G) ≤ (2/k^2 - 1/k^3) * cr(G)

The established bound is tight and cannot be replaced by any smaller constant than 1/k^2

Some results extend to rectilinear crossing numbers.

Abstract

The crossing number $cr(G)$ 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 $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to the rectilinear variants of the $k$-planar crossing number.