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arXiv · 1512.01335

On the Rectilinear Crossing Number of Complete Uniform Hypergraphs

Anurag Anshu·Rahul Gangopadhyay·Saswata Shannigrahi·Satyanarayana Vusirikala
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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 $\Theta(\frac{4^d}{\sqrt{d}}) {n \choose 2d}$.

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Anurag Anshu, Rahul Gangopadhyay, Saswata Shannigrahi, Satyanarayana Vusirikala. 2015-12-04. On the Rectilinear Crossing Number of Complete Uniform Hypergraphs. https://doi.org/10.1016/j.comgeo.2016.11.001

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