Discrete & Computational Geometry

, Volume 22, Issue 4, pp 633-642

First online:

Geometric Graphs with Few Disjoint Edges

  • G. TóthAffiliated withCourant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA toth@cims.nyu.edu
  • , P. ValtrAffiliated withDIMACS Center, Rutgers University, Piscataway, NJ 08855, USA

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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 k 3 (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.