The Euclidean MST-ratio for Bi-colored Lattices

Abstract

Given a finite set, $A\subseteq {\mathbb{R}}^2$, and a subset, $B\subseteq A$, the \emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A\setminus B$ divided by the length of the minimum spanning tree of $A$. The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$. Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than $1.25$.