Combinatorica

, Volume 17, Issue 3, pp 427–439

Graphs drawn with few crossings per edge

  • János Pach
  • Géza Tóth
Article

DOI: 10.1007/BF01215922

Cite this article as:
Pach, J. & Tóth, G. Combinatorica (1997) 17: 427. doi:10.1007/BF01215922

Abstract

We show that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv. Fork≤4, we establish a better bound, (k+3)(v−2), which is tight fork=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.

Mathematics Subject Classification (1991)

05C10 

Copyright information

© János Bolyai mathematical Society 1997

Authors and Affiliations

  • János Pach
    • 1
  • Géza Tóth
    • 1
  1. 1.Courant InstituteNYU and Hungarian Academy of SciencesHungary