Maximum Rectilinear Crossing Number of Uniform Hypergraphs

Abstract

We improve the lower bound on the d-dimensional rectilinear crossing number of the complete d-uniform hypergraph having 2d vertices to $\varOmega \left( \dfrac{(4\sqrt{2}/ 3^{3/4})^d}{d}\right)$ from $\varOmega (2^d \sqrt{d})$. We also establish that the 3-dimensional rectilinear crossing number of a complete 3-uniform hypergraph having $n \ge 9$ vertices is at least $\dfrac{43}{42}\genfrac(){0.0pt}0{n}{6}$. We prove that the maximum number of crossing pairs of hyperedges in a 4-dimensional rectilinear drawing of the complete 4-uniform hypergraph having n vertices is $13\genfrac(){0.0pt}0{n}{8}$. We also prove that among all 4-dimensional rectilinear drawings of a complete 4-uniform hypergraph having n vertices, the number of crossing pairs of hyperedges is maximized if all its vertices are placed at the vertices of a 4-dimensional neighborly polytope. Our result proves the conjecture by Anshu et al. [Anshu, Gangopadhyay, Shannigrahi, and Vusirikala, 2017] for $d=4$. We prove that the maximum d-dimensional rectilinear crossing number of a complete d-partite d-uniform balanced hypergraph is $(2^{d-1}-1){\genfrac(){0.0pt}0{n}{2}}^d$. We then prove that finding the maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph is NP-hard. We give a randomized scheme to create a d-dimensional rectilinear drawing of a d-uniform hypergraph H such that, in expectation the total number of crossing pairs of hyperedges is a constant fraction of the maximum d-dimensional rectilinear crossing number of H.