Discrete Mathematics & Theoretical Computer Science, Vol 15, No 1 (2013)

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The Erdős-Sós Conjecture for Geometric Graphs

Ruy Fabila-Monroy, Luis Felipe Barba, Dolores Lara, Jesús Leaños, Cynthia Rodríguez, Gelasio Salazar, Francisco Zaragoza

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 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when 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 completely solve the problem and prove that at least three edges must be removed.

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