Crossings in Grid Drawings

TL;DR

This paper establishes crossing number inequalities for geometric graphs on d-dimensional grids, providing bounds on crossings and counting non-crossing graphs, with specific results for 3D and higher dimensions.

Contribution

It introduces new crossing number inequalities for geometric graphs on d-dimensional grids and derives bounds on the number of non-crossing graphs, extending previous understanding.

Findings

In 3D, graphs with m >= 8N edges have ((m^2/n)\u2206(m/n)) crossings.

For d geq 4, the crossing bound simplifies to (m^2/n).

Maximum non-crossing graphs in d geq 4 are n^c n, with open bounds in 3D.

Abstract

We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has \Omega((m^2/n)\log(m/n)) crossings. In d-dimensions, with d >= 4, this bound becomes \Omega(m^2/n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some d-dimensional grid of volume N is n^\Theta(n). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2^\Omega(n) lower bound and the n^O(n) upper bound.