On the number of edges of separated multigraphs
(2108.11290)Abstract
We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n2\log n)$, 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 $\Omega\left(\frac{e3}{n2\log(e/n)}\right)$. This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.