Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Geom. 32(2), 177–194 (2004)) we study bottleneck version Euclidean bounded-degree spanning tree problem. A is a whose largest edge-length minimum, degree-K minimum. Let $$\beta _K$$ be supremum ratio to tree, over all finite point sets in plane. It known that _5=1$$ , it easy verify _2\geqslant 2$$ _3\geqslant \sqrt{2}$$ _4>1.175$$ . implied Hamiltonicity cube _2\leqslant 3$$ The degree-3 algorithm Ravi (25th Annual ACM Symposium on Theory Computing, pp. 438–447. ACM, New York (1993)) implies _3\leqslant Andersen Ras (Networks 68(4), 302–314 (2016)) showed _4\leqslant \sqrt{3}$$ We present following improved bounds: \sqrt{7}$$ As result, obtain better approximation algorithms for degree-4 trees. parts our proofs these bounds some structural properties minimum which are independent interest.