Disjoint edges in topological graphs and the tangled-thrackle conjecture

TL;DR

This paper proves that topological graphs avoiding certain disjoint edge configurations have linear edges, and applies this to confirm the tangled-thrackle conjecture, showing such graphs have at most linear edges in the plane.

Contribution

It establishes a linear bound on edges in topological graphs avoiding specific disjoint edge sets and confirms the tangled-thrackle conjecture.

Findings

Graphs with no two disjoint edge sets of size t have O(n) edges.

Confirmed the tangled-thrackle conjecture with an O(n) edge bound.

Provides structural insights into topological graph configurations.

Abstract

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i\'c, and T\'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges.