Empty Triangles in Generalized Twisted Drawings of $K_n$

TL;DR

This paper proves that all generalized twisted drawings of the complete graph $K_n$ contain exactly 2n-4 empty triangles, advancing the understanding of geometric properties of such graph drawings.

Contribution

It establishes the exact number of empty triangles in generalized twisted drawings of $K_n$, a key step towards a broader conjecture for all simple drawings.

Findings

All generalized twisted drawings of $K_n$ have exactly 2n-4 empty triangles.

The result supports the conjecture for all simple drawings of $K_n$.

Progress in understanding geometric configurations in graph drawings.

Abstract

Simple drawings are drawings of graphs in the plane or on the sphere such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared endpoint). Simple drawings are generalized twisted if there is a point $O$ such that every ray emanating from $O$ crosses every edge of the drawing at most once and there is a ray emanating from $O$ which crosses every edge exactly once. We show that all generalized twisted drawings of $K_n$ contain exactly $2n-4$ empty triangles, by this making a substantial step towards proving the conjecture that this is the case for every simple drawing of $K_n$.