Note on disjoint faces in simple topological graphs

TL;DR

This paper proves that complete simple topological graphs with n vertices have at least Omega(n) disjoint 4-faces, leading to bounds on the minimal area of such faces, and explores related problems for k-faces.

Contribution

It establishes a lower bound on the number of disjoint 4-faces in complete simple topological graphs, improving previous results and providing tight examples.

Findings

At least Omega(n) disjoint 4-faces in complete simple topological graphs.

Existence of a 4-face with area at most O(1/n) in such graphs.

Construction of examples showing the bounds are asymptotically tight.

Abstract

We prove that every $n$-vertex complete simple topological graph generates at least $\Omega(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$.