On the Maximum Number of Crossings in Star-Simple Drawings of $K_n$ with No Empty Lens

TL;DR

This paper investigates star-simple drawings of complete graphs without empty lenses, establishing an upper bound of 3 times factorial of (n-4) on pairwise crossings, leading to a finite total crossing bound of n!.

Contribution

It introduces a new upper bound on the maximum crossings between any two edges in star-simple drawings without empty lenses of K_n.

Findings

Maximum pairwise crossings bounded by 3((n-4)!)

Total crossings in such drawings are bounded by n!

Crossings are finite and well-controlled in this setting

Abstract

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of $K_n$ with no empty lens. In this setting we prove an upper bound of $3((n-4)!)$ on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite and upper bounded by $n!$.