Various worst-case ratios of minimum weight bounded-degree spanning
trees for different degree bounds are still open, in particular
comparing τ_{k} to the weight τ of a minimum spanning tree.
[FKK+97] conjecture
τ_{3}/τ≤1.103...,
τ_{4}/τ≤1.035... for Euclidean distances in the plane,
and
τ_{4}/τ≤1.25 for Manhattan distances in the plane,
and give matching lower bounds.
[KRY96] show that for Euclidean distances,
τ_{4}/τ≤1.25 and
τ_{3}/τ≤1.5 in the plane,
and
τ_{3}/τ≤1.66... in arbitrary dimensions.
The first two of these bounds were improved to
τ_{4}/τ≤1.143 and
τ_{3}/τ≤1.402
by [Cha03].
Now settled by NP-hard proof in [FH09].