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, Pach and Tóth (2009).
@InProceedings{biniaz:LIPIcs.SoCG.2022.16, author = {Biniaz, Ahmad}, title = {{Acute Tours in the Plane}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {16:1--16:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.16}, URN = {urn:nbn:de:0030-drops-160240}, doi = {10.4230/LIPIcs.SoCG.2022.16}, annote = {Keywords: planar points, acute tour, Hamiltonian cycle, equitable partition} }
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