Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

TL;DR

This paper investigates the geometric properties of complete simple topological graphs, establishing lower bounds on disjoint faces and exploring variants of the Heilbronn triangle problem related to area minimization.

Contribution

It proves that every complete simple topological graph on n vertices has at least Omega(n^{1/3}) disjoint 4-faces, linking face disjointness to area bounds and Heilbronn problems.

Findings

Every complete n-vertex simple topological graph has at least Omega(n^{1/3}) disjoint 4-faces.

Any such graph in the unit square contains a 4-face with area O(n^{-1/3}).

The paper explores a Z_2 variant of the Heilbronn triangle problem.

Abstract

Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $\Omega(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. Finally, we investigate a $\mathbb Z_2$ variant of Heilbronn triangle problem.