The Erd\H{o}s-S\'os Conjecture for Geometric Graphs

TL;DR

This paper investigates the minimum edges removal needed in complete geometric graphs to eliminate certain trees as planar subgraphs, providing bounds and special cases for different values of k and point configurations.

Contribution

It establishes new bounds on the function f(n,k) for geometric graphs and explores specific cases like when k=n and points are in convex position.

Findings

Bounds on f(n,k) for general geometric graphs

Exact bounds for the case k=n

Minimum edges to remove in convex position cases

Abstract

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that $(1/2)\frac{n^2}{k-1}-\frac{n}{2}\le f(n,k) \le 2 \frac{n(n-2)}{k-2}$. For the case when $k=n$, we show that $2 \le f(n,n) \le 3$. For the case when $k=n$ and $G$ is a geometric graph on a set of points in convex position, we show that at least three edges must be removed.