Discrete & Computational Geometry

, Volume 22, Issue 4, pp 633–642

Geometric Graphs with Few Disjoint Edges

  • G. Tóth
  • P. Valtr

DOI: 10.1007/PL00009482

Cite this article as:
Tóth, G. & Valtr, P. Discrete Comput Geom (1999) 22: 633. doi:10.1007/PL00009482


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.

Copyright information

© 1998 Springer-Verlag New York Inc.

Authors and Affiliations

  • G. Tóth
    • 1
  • P. Valtr
    • 2
  1. 1.Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA toth@cims.nyu.edu US
  2. 2.DIMACS Center, Rutgers University, Piscataway, NJ 08855, USA US
  3. 3.Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic valtr@kam.mff.cuni.czCZ