Packing Plane Spanning Trees into a Point Set

TL;DR

This paper proves that at least one-third of the points in a plane can be covered by non-overlapping plane spanning trees, significantly improving previous bounds, and explores properties of point set centers in higher dimensions.

Contribution

It establishes a new lower bound of  plane spanning trees in a point set, advancing combinatorial geometry understanding.

Findings

At least  trees can be packed into a point set.

Improved lower bound from  to .

Center of a point set in -dimensional space is either 0- or d-dimensional.

Abstract

Let $P$ be a set of $n$ points in the plane in general position. We show that at least $\lfloor n/3\rfloor$ plane spanning trees can be packed into the complete geometric graph on $P$. This improves the previous best known lower bound $\Omega\left(\sqrt{n}\right)$. Towards our proof of this lower bound we show that the center of a set of points, in the $d$-dimensional space in general position, is of dimension either $0$ or $d$.