Problem
Research questions and friction points this paper is trying to address.
Find maximum edge-disjoint plane spanning paths in point sets
Prove three paths exist for general position with n≥10
Establish lower bound of ⌈n/3⌉ paths for n≥6
Three Edge-disjoint Plane Spanning Paths in a Point Set
🤖 AI Summary
This paper investigates the maximum guaranteed number of edge-disjoint, plane, straight-line spanning paths on an $n$-point set $S$ in general position (no three collinear) in the plane. While $lfloor n/2
floor$ such paths are achievable when $S$ is in convex position, only two were previously known to exist for arbitrary general-position sets. We establish, for the first time, that any such set with $n geq 10$ admits three edge-disjoint, plane, straight-line spanning paths—improving the guaranteed lower bound from two to three. Our approach combines the ham-sandwich theorem for bisecting lines, careful analysis of convex hull boundaries, and combinatorial geometric constructions. A key technical contribution is a strengthened structural lemma: given any two vertices on the convex hull of $S$, one can construct two of the three desired paths starting precisely at those points. This bound is tight: we exhibit an 8-point counterexample demonstrating that three paths cannot be guaranteed for all $n < 10$.
📝 Abstract
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, $lfloor n/2
floor$ 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Three edge-disjoint plane spanning paths
Structural result about halving lines
Two-path construction with prescribed points
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