Maximum Rectilinear Crossing Number of Uniform Hypergraphs

We improve the lower bound on the d -dimensional rectilinear crossing number of the complete d -uniform hypergraph having 2 d vertices to ( (4√(2)/ 3^3/4)^dd) from (2^d √(d)) . We also establish that the 3-dimensional rectilinear crossing number of a complete 3-uniform hypergraph having n ≥ 9 vertices is at least 4342()0.0pt0n6 . 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()0.0pt0n8 . 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)()0.0pt0n2^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 .