Rectilinear crossing number of uniform hypergraphs

Abstract

In this thesis, we study the d-dimensional rectilinear drawings of d-uniform hypergraphs in which each hyperedge contains exactly d vertices. A d-dimensional rectilinear drawing of a d-uniform hypergraph is a drawing of the hypergraph in Rd when its vertices are placed as points in general position and its hyperedges are drawn as the convex hulls of the corresponding d points. In such a drawing, a pair of hyperedges forms a crossing if they are vertex disjoint and contain a common point in their relative interiors. A special kind of d-dimensional rectilinear drawing of a d-uniform hypergraph is known as a d-dimensional convex drawing of it when its vertices are placed as points in general as well as in convex position in Rd. The d-dimensional rectilinear crossing number of a d-uniform hypergraph is the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of it. Similarly, the d-dimensional convex crossing number of a d-uniform hypergraph is the minimum number of crossing pairs of hyperedges among all d-dimensional convex drawings of it. We study two types of uniform hypergraphs in this thesis, namely, the complete d-uniform hypergraphs and the complete balanced d-partite d-uniform hypergraphs. We summarise the main results of this thesis as follows.

We prove that the d-dimensional rectilinear crossing number of a complete d-uniform hypergraph having n vertices is Ω(2^d⋅√d)(█(n@2d)).

We prove that any 3-dimensional convex drawing of a complete 3-uniform hypergraph with n vertices contains 3(n¦6) crossing pairs of hyperedges.

We prove that there exist θ(4^d/√d)(█(n@2d)) crossing pairs of hyperedges in the d-dimensional rectilinear drawing of a complete d-uniform hypergraph having n vertices when all its vertices are placed over the d-dimensional moment curve.

We prove that the d-dimensional rectilinear crossing number of a complete balanced d-partite d-uniform hypergraph having nd vertices is Ω(2^d ) (n/2)^d (((n-1))/2)^d.

We also study the properties of different types of d-dimensional rectilinear drawings and d-dimensional convex drawings of the complete d-uniform hypergraph having 2d vertices by exploiting its relations with convex polytopes and k-sets.