On the number of edges of separated multigraphs

Abstract

AbstractWe prove that the number of edges of a multigraph with vertices is at most , provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi, and Leighton, if has edges, in any drawing of with the above property, the number of crossings is . This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.