The number of crossings in multigraphs with no empty lens

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 T\'oth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton (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 $\alpha e^3/n^2$, for a suitable constant $\alpha>0$. 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 $\alpha e^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Crossings between non-homotopic edges

We call a multigraph {\em non-homotopic} if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on $n>1$ vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with $n$ vertices and $m>4n$ edges is larger than $c\frac{m^2}{n}$ for some constant $c>0$, and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as $n$ is fixed and $m$ tends to infinity.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020