Disjoint edges in complete topological graphs

TL;DR

This paper proves that every complete simple topological graph with n vertices contains at least on the order of n^{1/3} pairwise disjoint edges, and provides a polynomial-time method to find them.

Contribution

It confirms a conjecture by Pach and Tóth by establishing a lower bound on disjoint edges and offering an efficient algorithm to identify such edges.

Findings

At least Omega(n^{1/3}) disjoint edges exist in any complete simple topological graph.

Disjoint edges can be found in polynomial time.

The result confirms a conjecture of Pach and Tóth.

Abstract

It is shown that every complete n-vertex simple topological graph has at least Omega(n^{1/3}) pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and T\'oth.