On the number of edges of separated multigraphs

TL;DR

This paper establishes an upper bound on the edges of certain multigraphs with crossing constraints and extends the Crossing Lemma, providing tight bounds up to a logarithmic factor for graphs with many edges.

Contribution

It introduces a bound on multigraph edges under crossing restrictions and extends the Crossing Lemma to a broader class of graphs, answering an open question.

Findings

Maximum edges of multigraphs is O(n^2 log n) under given conditions.

Number of crossings in such graphs is Omega(e^3 / (n^2 log(e/n))).

Results are tight up to a logarithmic factor.

Abstract

We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$ contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chv\'atal, Newborn, Szemer\'edi and Leighton, if $G$ has $e \geq 4n$ edges, in any drawing of $G$ with the above property, the number of crossings is $\Omega\left(\frac{e^3}{n^2\log(e/n)}\right)$. This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.