## 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, March 28, 2013, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Heuna Kim (Korea Advanced Institute of Science and Technology)

## On the Numbers of Edges of a Fan-Crossing Free Graph

A graph drawn in the plane with n vertices is fan-crossing free if there is no triple of edges e, f, and g, such that e and f have a common endpoint and g crosses both e and f. Fan-crossing free graphs arise, for instance, in graph drawing. Humans can read graph drawings in which edges cross at right angles well. Unfortunately, there is previous research showing that only small graphs can be drawn this way: A straight-line drawing of a graph using only right-angle crossings has at most 4n − 10 edges. Since such a graph is fan-crossing free, this led to the question: "what is the maximum number of edges of a fan-crossing free graph on n vertices?"

We answer this question by showing that a fan-crossing free graph has at most 4n − 8 edges (and at most 4n − 9 edges with straight-line drawings). We generalize our result to graphs without radial (k, 1)-grids; that is, sets of k edges all incident to a common endpoint that are all crossed by another edge e. Finally, we give a very general bound for a monotone graph property.

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