Acute Tours in the Plane

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 π /2 . Our proof is constructive and suggests a simple O(nlog n) -time algorithm for finding such a tour. The previous best-known upper bound on the angle is 2π /3 , and it is due to Dumitrescu et al. (Electron. J. Comb. 19 (2), # P31 (2012)).