On the Rectilinear Crossing Number of Complete Uniform Hypergraphs

TL;DR

This paper improves the lower bound on the rectilinear crossing number for complete d-uniform hypergraphs in d-dimensional space and analyzes the crossing number when vertices are on a moment curve.

Contribution

It provides a tighter lower bound for the crossing number and characterizes the crossing number for hypergraphs with vertices on a moment curve.

Findings

Lower bound improved to  2^d {n  2d}

Crossing number for vertices on moment curve is  4^d /   2d

Results advance understanding of hypergraph crossing numbers in high dimensions

Abstract

In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the $d$-dimensional rectilinear drawing of a $d$-uniform hypergraph is known as the $d$-dimensional rectilinear crossing number of the hypergraph. The currently best-known lower bound on the $d$-dimensional rectilinear crossing number of a complete $d$-uniform hypergraph with $n$ vertices in general position in $\mathbb{R}^d$ is $\Omega(\frac{2^d}{\sqrt{d}} \log d) {n \choose 2d}$. In this paper, we improve this lower bound to $\Omega(2^d) {n \choose 2d}$. We also consider the special case when all the vertices of a $d$-uniform hypergraph are placed on the $d$-dimensional moment curve. For such complete $d$-uniform hypergraphs with $n$ vertices, we show that the number of pairwise crossing hyperedges is $\Theta(\frac{4^d}{\sqrt{d}}) {n \choose 2d}$.