  Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG
2003) 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 $\beta_2\ge 2$, $\beta_3\ge \sqrt{2}$, and
$\beta_4>1.175$.
  It is implied by the Hamiltonicity of the cube of the bottleneck spanning
tree that $\beta_2\le 3$. The degree-3 spanning tree algorithm of Ravi et al.
(STOC 1993) implies that $\beta_3\le 2$. Andersen and Ras (Networks,
68(4):302-314, 2016) showed that $\beta_4\le \sqrt{3}$. We present the
following improved bounds: $\beta_2\ge\sqrt{7}$, $\beta_3\le \sqrt{3}$, and
$\beta_4\le \sqrt{2}$. 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.
