Non-Homotopic Drawings of Multigraphs

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13n(k+1)}$, which is a substantial improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2\left(\genfrac{}{}{0px}{}{2n}{k+1}\right)$ and the number of crossings is $\mathrm{\Omega }\left(\frac{m^{2+1\mathrm{/}k}}{n^{1+1\mathrm{/}k}}\right)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.