On the number of edges of separated multigraphs
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We prove that the number of edges of a multigraph G with n vertices is at most O(n^2log 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, Chvátal, Newborn, Szemerédi and Leighton, if G has e ≥ 4n edges, in any drawing of G with the above property, the number of crossings is Ω(e^3/n^2log(e/n)). This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.
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