New bounds on the maximum number of edges in $k$-quasi-planar graphs

TL;DR

This paper improves the upper bounds on the maximum number of edges in $k$-quasi-planar graphs, advancing understanding of their structural limits and addressing longstanding conjectures in topological graph theory.

Contribution

It provides tighter upper bounds on edges in $k$-quasi-planar graphs, especially for those with bounded edge intersections, refining previous results by Fox, Pach, and Suk.

Findings

Improved upper bound to $2^{ ext{α}(n)^c} n ext{log} n$ for certain $k$-quasi-planar graphs.

Established an $O(n ext{log} n)$ bound for graphs with edges intersecting at most once.

Extended bounds to graphs with bounded pairwise edge intersections.

Abstract

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox and Pach showed that every $k$-quasi-planar graph with $n$ vertices has at most $n(\log n)^{O(\log k)}$ edges. We improve this upper bound to $2^{\alpha(n)^c}n\log n$, where $\alpha(n)$ denotes the inverse Ackermann function and $c$ depends only on $k$, for $k$-quasi-planar graphs in which any two edges intersect in a bounded number of points. We also show that every $k$-quasi-planar graph with $n$ vertices in which any two edges have at most one point in common has at most $O(n\log n)$ edges. This improves the previously known upper bound of $2^{\alpha(n)^c}n\log n$ obtained by Fox, Pach, and Suk.