On the maximum number of edges of outer k-planar graphs

We study the maximum number of straight-line segments connecting $n$ points in convex position in the plane, so that each segment intersects at most $k$ others. This question can also be framed as the maximum number of edges of an outer $k$-planar graph on $n$ vertices. We outline several approaches to tackle the problem with the best approach yielding an upper bound of $(\sqrt{2}+\varepsilon)\sqrt{k}n$ edges (with $\varepsilon \rightarrow 0$ for sufficiently large $k$). We further investigate the case where the points are arbitrarily bicolored and segments always connect two different colors (i.e., the corresponding graph has to be bipartite). To this end, we also consider the maximum cut problem for the circulant graph $C_n^{1,2,\dots,r}$ which might be of independent interest.