The number of crossings in multigraphs with no empty lens

TL;DR

This paper investigates the minimum number of crossings in multigraphs with no empty lens, extending previous crossing lemmas to cases where nonparallel edges can cross multiple times, establishing tight bounds.

Contribution

It extends the crossing lemma to multigraphs allowing multiple crossings between nonparallel edges, providing tight bounds on the minimum number of crossings.

Findings

For multigraphs with no empty lens, the crossing number is at least proportional to e^{2.5}/n^{1.5}.

The derived bounds are tight and cannot be improved in order of magnitude.

The results generalize previous crossing lemmas to more complex multigraph configurations.

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 T\'oth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of computing series, 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.