Maximum Rectilinear Crossing Number of Uniform Hypergraphs

In this paper, we study the maximum $d$-dimensional rectilinear crossing number problem of $d$-uniform hypergraphs. In \cite{AGS}, Anshu et al. conjectured that among all $d$-dimensional rectilinear drawings of a complete $d$-uniform hypergraph having $n$ vertices, the number of crossing pairs of hyperedges is maximized if all of its vertices are placed on the $d$-dimensional moment curve, and proved it for $d=3$; with it being trivially true for $d = 2$, since any convex drawing of the complete graph $K_n$ produces $n \choose 4$ pairs of crossing edges. In this paper, we prove that their conjecture is valid for $d=4$ by using Gale transform. We also prove that the maximum $d$-dimensional rectilinear crossing number of a complete $d$-partite $d$-uniform balanced hypergraph is $(2^{d-1}-1){n \choose 2}^d$, where $n$ denotes the number of vertices in each part. We then prove that finding the maximum $d$-dimensional rectilinear crossing number of an arbitrary $d$-uniform hypergraph is NP-hard. For a $d$-uniform hypergraph $H$, we also give a randomized scheme to create a $d$-dimensional rectilinear drawing of $H$ which produces the number of crossing pair of hyperedges up to a constant factor of the maximum $d$-dimensional rectilinear crossing number of $H$.