Acute Tours in the Plane

TL;DR

This paper proves a long-standing conjecture that large point sets in the plane can be connected by a Hamiltonian cycle with angles between edges at most 90 degrees, providing a constructive $O(n ext{log} n)$ algorithm.

Contribution

It confirms Fekete and Woeginger's conjecture for large even point sets and introduces a simple, efficient algorithm to find such tours.

Findings

Confirmed the conjecture for sufficiently large even n

Developed an $O(n ext{log} n)$-time constructive algorithm

Improved the upper bound on the angle from $2 ext{pi}/3$ to $ ext{pi}/2$

Abstract

We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number $n$, every set of $n$ points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line edges such that the angle between any two consecutive edges is at most $\pi/2$. Our proof is constructive and suggests a simple $O(n\log n)$-time algorithm for finding such a tour. The previous best-known upper bound on the angle is $2\pi/3$, and it is due to Dumitrescu, Pach and T\'oth (2009).