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

受到Khuller等人的开创性著作的启发。（SIAM J. COMPUT。25（2），355-368（1996））和陈（离散COMPUT。的Geom。32（2），177-194（2004））研究了欧几里德的瓶颈版本界度跨越树问题。瓶颈生成树是最大边缘长度最小的生成树，而瓶颈度K生成树是最大边缘长度最小的度K生成树。令\（\ beta _K \）为在欧几里得平面上所有有限点集上，瓶颈度K生成树的最大边缘长度与瓶颈生成树的最大边缘长度的最大比。已知\（\ beta _5 = 1 \），并且很容易验证\（\ beta _2 \ geqslant 2 \），\（\ beta _3 \ geqslant \ sqrt {2} \）和\（\ beta _4> 1.175 \）。瓶颈生成树的立方的汉密尔顿性暗示\（\ beta _2 \ leqslant 3 \）。Ravi等人的3度生成树算法。（第25届ACM计算理论年度研讨会，第438-447页。ACM，纽约（1993年））暗示\（\ beta _3 \ leqslant 2 \）。Andersen和Ras（网络68（4），302–314（2016））显示\（\ beta _4 \ leqslant \ sqrt {3} \）。我们提供以下改进的边界：\（\ beta _2 \ geqslant \ sqrt {7} \），\（\ beta _3 \ leqslant \ sqrt {3} \）和\（\ beta _4 \ leqslant \ sqrt {2} \）。结果，我们为欧几里得瓶颈度3和度4生成树获得了更好的近似算法。作为我们对这些边界的证明的一部分，我们提出了具有独立利益的欧几里得最小生成树的一些结构特性。