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

TL;DR

This paper investigates the maximum number of edges in outer $k$-planar graphs with points in convex position, providing upper bounds and exploring bipartite cases and related maximum cut problems.

Contribution

It introduces new upper bounds for the maximum edges in outer $k$-planar graphs and examines bipartite and circulant graph cases.

Findings

Upper bound of $( ext{sqrt{2}}+ ext{varepsilon}) ext{sqrt{k}}n$ edges for large $k$

Analysis of bipartite outer $k$-planar graphs

Study of maximum cut in circulant graphs

Abstract

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.