Construction of Locally Plane Graphs with Many Edges
A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most k in a geometric graph G is self-intersecting, we call Gk-locally plane. The main result of this chapter is a construction of k-locally plane graphs with a superlinear number of edges. For the proof, we develop randomized thinning procedures for edge-colored bipartite (abstract) graphs that can be applied to other problems as well.
The author’s research was partially supported by NSERC Grant 329527 and by OTKA Grants T-046234, AT048826, and NK-62321.
- 3.S. Avital, H. Hanani, Graphs. Gilyonot Lematematika 3, 2–8 (1966) [in Hebrew]Google Scholar
- 5.J. Pach, Geometric graph theory, in Surveys in Combinatorics, 1999, ed. by J. Lamb, D. Preece. London Mathematical Society Lecture Notes, vol. 267 (Cambridge University Press, Cambridge, 1999), pp. 167–200Google Scholar
- 6.J. Pach, Geometric graph theory, in Handbook of Discrete and Computational Geometry, 2nd edition, ed. by J. Goodman, J. O’Rourke (Chapman & Hall/CRC, Boca Raton, FL, 2004) ( Chapter 10)
- 8.J. Pach, R. Radoičić, G. Tóth, Relaxing planarity for topological graphs, in Discrete and Computational Geometry, ed. by J. Akiyama, M. Kano. Lecture Notes in Computer Science, vol. 2866 (Springer, Berlin, 2003), pp. 221–232Google Scholar
- 10.R. Pinchasi, R. Radoičić, On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342 (American Mathematical Society, Providence, RI, 2004), pp. 233–243Google Scholar
- 11.P. Valtr, Graph drawing with no k pairwise crossing edges, in Graph Drawing (Rome, 1997). Lecture Notes in Computer Science, vol. 1353 (Springer-Verlag, Berlin, 1997), pp. 205–218Google Scholar