The number of crossings in multigraphs with no empty lens
Abstract
Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton (1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in $G$ is at least $\alpha e^3/n^2$, for a suitable constant $\alpha>0$. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in $G$ is at least $\alpha e^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1808.10480
 Bibcode:
 2018arXiv180810480K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Geometry
 EPrint:
 Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)