On topological graphs with at most four crossings per edge

TL;DR

This paper proves a bound on the maximum edges in topological graphs with at most four crossings per edge, improving the crossing lemma bound and confirming the Albertson conjecture for graphs with chromatic number up to 18.

Contribution

It establishes a new upper bound on edges for graphs with limited crossings, advances the crossing lemma, and extends the validity of the Albertson conjecture.

Findings

Graphs with at most four crossings per edge have at most 6n-12 edges.

The crossing number bound is improved to c(m^3/n^2) with c > 1/29.

Albertson conjecture holds for graphs with chromatic number up to 18.

Abstract

We show that if a graph $G$ with $n \geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\v{c}i\'{c}, Tardos, and T\'oth, and yields a better bound for the famous Crossing Lemma: The crossing number, $\mbox{cr}(G)$, of a (not too sparse) graph $G$ with $n$ vertices and $m$ edges is at least $c\frac{m^3}{n^2}$, where $c > 1/29$. This bound is known to be tight, apart from the constant $c$ for which the previous best lower bound was $1/31.1$. As another corollary we obtain some progress on the Albertson conjecture: Albertson conjectured that if the chromatic number of a graph $G$ is $r$, then $\mbox{cr}(G) \geq \mbox{cr}(K_r)$. This was verified by Albertson, Cranston, and Fox for $r \leq 12$, and for $r \leq 16$ by Bar\'at and T\'oth. Our results imply that Albertson conjecture holds for $r \leq 18$.