We study the following problem: Given a set S of n points in the plane,
how many edge-disjoint plane straight-line spanning paths of S can one draw?
A well known result is that when the n points are in convex position,
⌊n/2⌋ such paths always exist, but when the points of S are in
general position the only known construction gives rise to two edge-disjoint
plane straight-line spanning paths. In this paper, we show that for any set S
of at least ten points, no three of which are collinear, one can draw at least
three edge-disjoint plane straight-line spanning paths of~S. Our proof is
based on a structural theorem on halving lines of point configurations and a
strengthening of the theorem about two spanning paths, which we find
interesting in its own right: if S has at least six points, and we prescribe
any two points on the boundary of its convex hull, then the set contains two
edge-disjoint plane spanning paths starting at the prescribed points.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023