## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, July 16, 2015, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Andres Ruiz-Vargas (EPF Lausanne)

## Disjoint edges and empty triangles in topological graphs

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. In this talk, we present a lemma for constructing plane subgraphs in simple topological graphs. We will discuss the following two applications of this lemma.
Let $G$ be a complete simple topological graph on $n$ vertices. The three edges induced by any triplet of vertices in $G$ form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that $G$ has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $2n/3$. We settle Harborth's conjecture in the affirmative.
We also sketch a proof that every complete simple topological graph on $n$ vertices contains $\Omega(n^\frac 12)$ pairwise disjoint edges.

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