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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,$\lfloor n/2\rfloor$ such paths always exist, but when the points of $S$ are ingeneral position the only known construction gives rise to two edge-disjointplane 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 leastthree edge-disjoint plane straight-line spanning paths of~$S$. Our proof isbased on a structural theorem on halving lines of point configurations and astrengthening of the theorem about two spanning paths, which we findinteresting in its own right: if $S$ has at least six points, and we prescribeany two points on the boundary of its convex hull, then the set contains twoedge-disjoint plane spanning paths starting at the prescribed points.

Three Edge-disjoint Plane Spanning Paths in a Point Set