The Erdős-Sós conjecture for geometric graphs

Combinatoric

The Erdős-Sós Conjecture for Geometric Graphs

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 ( ) 2 1 n n n(n−2) − ≤ f(n, k) ≤ 2. For the case when 2 k−1 2 k−2 k = n, we show that 2 ≤ f(n, n) ≤ 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.