## Geometric Graphs with Few Disjoint Edges

### Authors: Géza Tóth and Pavel Valtr

ABSTRACT

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points.

Improving a result of Pach and T\"or\H ocsik, we show that a geometric graph on $n$ vertices with no $k+1$ pairwise disjoint edges has at most $k^3(n+1)$ edges. On the other hand, we construct geometric graphs with $n$ vertices and approximately ${3\over 2}(k-1)n$ edges, containing no $k+1$ pairwise disjoint edges.

We also improve both the lower and upper bounds of Goddard, Katchalski and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1998/98-22.ps.gz