Geometric Graphs with Few Disjoint Edges
- Cite this article as:
- Tóth, G. & Valtr, P. Discrete Comput Geom (1999) 22: 633. doi:10.1007/PL00009482
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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 Törőcsik, we show that a geometric graph on n vertices with no k+1 pairwise disjoint edges has at most k3(n+1) edges. On the other hand, we construct geometric graphs with n vertices and approximately (3/2)(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.