The number of crossings in multigraphs with no empty lens - Kaufmann, Michael; Pach, János; Tóth, Géza; Ueckerdt, Torsten (Journal of Graph Algorithms and Applications, 2021)
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 Tóth [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 $α e^3/n^2$, for a suitable constant $α>0$. 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 $α e^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.