Computational search of small point sets with small rectilinear crossing number

TL;DR

This paper improves the upper bound on the rectilinear crossing number of complete graphs by constructing better small point sets and using known methods to extend these to larger graphs.

Contribution

It introduces a new heuristic for finding small point sets with low crossing numbers and improves the upper bound on rs(K_n) for complete graphs.

Findings

New rectilinear drawings for small n

Improved upper bound on rs(K_n)

Effective heuristic for small point set search

Abstract

Let $\crs(K_n)$ be the minimum number of crossings over all rectilinear drawings of the complete graph on $n$ vertices on the plane. In this paper we prove that $\crs(K_n) < 0.380473\binom{n}{4}+\Theta(n^3)$; improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of $K_n$ for small values of $n$, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The "small" sets where found using a simple heuristic detailed in this paper.