On the rectilinear crossing number of complete uniform hypergraphs

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 R d is Ω( 2 d √ d log d) n 2d . In this paper, we improve this lower bound to Ω(2 d ) n 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 crossing pairs of hyperedges is Θ( 4 d √ d ) n 2d .

k-Sets and rectilinear crossings in complete uniform hypergraphs

“…Two non-empty convex sets are said to be properly separated in R d if they lie in the opposite closed half-spaces created by a (d − 1)dimensional hyperplane and both of them are not contained in the hyperplane. The proof of the following lemma, which is used in the proofs of all four theorems, is the same proof mentioned in [4] for the special case u = v = d − 1. For the sake of completeness, we repeat its proof in full generality.…”

“…A u-simplex and a v-simplex are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors [5]. As a result, a pair of hyperedges in a d-dimensional rectilinear drawing of K d n are said to be crossing if they do not have a common vertex and contain a common point in their relative interiors [3,4,5]. The d-dimensional rectilinear crossing number of K d n , denoted by cr d (K d n ), is defined as the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of K d n [3,4].…”

“…As a result, a pair of hyperedges in a d-dimensional rectilinear drawing of K d n are said to be crossing if they do not have a common vertex and contain a common point in their relative interiors [3,4,5]. The d-dimensional rectilinear crossing number of K d n , denoted by cr d (K d n ), is defined as the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of K d n [3,4]. Since the crossing pairs of hyperedges formed by a set containing 2d vertices of K d n are distinct from the crossing pairs of hyperedges formed by another set of 2d vertices, it can be observed that cr d (K d n ) ≥ cr d (K d 2d ) n 2d [3].…”

“…Since the crossing pairs of hyperedges formed by a set containing 2d vertices of K d n are distinct from the crossing pairs of hyperedges formed by another set of 2d vertices, it can be observed that cr d (K d n ) ≥ cr d (K d 2d ) n 2d [3]. The bestknown lower bound on cr d (K d 2d ) is Ω(2 d ) [4]. Anshu et al [4] also studied a d-dimensional rectilinear drawing of K d 2d where all its vertices are placed on the d-dimensional moment curve γ = {(a, a 2 , .…”

“…it is a collection of d + 5 vectors in R4 . Lemma 4 implies that there exists a linear hyperplane h that partitions D(V ′ ) in such a way that one of the open half-spaces created by h contains exactly one vector of D(V ′ ).…”