Generalized Thrackles and Geometric Graphs in ${\mathbb{R}}^3$ with No Pair of Strongly Avoiding Edges

Abstract

We define the notion of a geometric graph in ${\mathbb{R}}^3$. This is a graph drawn in ${\mathbb{R}}^3$ with its vertices drawn as points and its edges as straight line segments connecting corresponding points. We call two disjoint edges of G strongly avoiding if there exists an orthogonal projection of $\mathbb{R}^3$ to a two dimensional plane H such that the projections of the two edges on H are contained in two different rays, respectively, with a common apex that create a non-acute angle. We show that a geometric graph on n vertices in $\mathbb{R}^3$ with no pair of strongly avoiding edges has at most 2n − 2 edges. As a consequence we get a new proof to Vázsonyi’s conjecture about the maximum number of diameters in a set of n points in ${\mathbb{R}}^3$.