An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants

TL;DR

This paper discusses an ongoing project aimed at improving asymptotic upper bounds for the rectilinear and pseudolinear crossing numbers of complete graphs, which measure the minimum crossings in specific graph drawings.

Contribution

It introduces a continuous effort to refine upper bounds on crossing numbers for complete graphs in rectilinear and pseudolinear drawings.

Findings

Progress towards tighter asymptotic bounds

Development of new drawing techniques for complete graphs

Potential for improved crossing number estimates

Abstract

A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of edges crosses precisely once. A special case are {\it rectilinear} drawings where the edges of the graph are drawn as straight line segments. The rectilinear (pseudolinear) crossing number of a graph is the minimum number of pairs of edges of the graph that cross in any of its rectilinear (pseudolinear) drawings. In this paper we describe an ongoing project to continuously obtain better asymptotic upper bounds on the rectilinear and pseudolinear crossing number of the complete graph $K_n$.