Point sets that minimize $(\le k)$-edges, 3-decomposable drawings, and the rectilinear crossing number of $K_{30}$

TL;DR

This paper explores the relationship between point set properties that minimize crossings in geometric drawings of complete graphs, establishing a link between edge counts and 3-decomposability, and applies this to determine the crossing number of K_{30}.

Contribution

It proves that point sets with specific edge properties are necessarily 3-decomposable and determines the rectilinear crossing number of K_{30}.

Findings

Point sets with exactly 3inom{k+2}{2} ( extless k) edges are 3-decomposable.

The rectilinear crossing number of K_{30} is 9726.

A tight relationship between edge counts and 3-decomposability is established.

Abstract

There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le k < n/3$. Second, all such drawings have the $n$ points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are tightly related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\le k)$-edges for all $0\le k < n/3$, is 3-decomposable. As an application, we prove that the rectilinear crossing number of $K_{30}$ is 9726.