Improved Bounds for Covering Paths and Trees in the Plane

A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane can be covered by a noncrossing path with at most $π(n)$ edges. Let $τ(n)$ be the analogous number for noncrossing covering trees. Dumitrescu, Gerbner, Keszegh, and Tóth (Discrete & Computational Geometry, 2014) established the following inequalities: \[\frac{5n}{9} - O(1) < π(n) < \left(1-\frac{1}{601080391}\right)n, \quad\text{and} \quad\frac{9n}{17} - O(1) < τ(n)\leqslant \left\lfloor\frac{5n}{6}\right\rfloor.\] We report the following improved upper bounds: \[π(n)\leqslant \left(1-\frac{1}{22}\right)n, \quad\text{and}\quad τ(n)\leqslant \left\lceil\frac{4n}{5}\right\rceil.\] In the same context we study rainbow polygons. For a set of colored points in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color in its interior or on its boundary. Let $ρ(k)$ be the minimum number such that every $k$-colored point set in the plane admits a perfect rainbow polygon of size $ρ(k)$. Flores-Peñaloza, Kano, Martínez-Sandoval, Orden, Tejel, Tóth, Urrutia, and Vogtenhuber (Discrete Mathematics, 2021) proved that $20k/19 - O(1) <ρ(k) < 10k/7 + O(1).$ We report the improved upper bound $ρ(k)< 7k/5 + O(1)$. To obtain the improved bounds we present simple $O(n\log n)$-time algorithms that achieve paths, trees, and polygons with our desired number of edges.