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 $1.25$.

Figures (9)

• Figure 1: Left: a portion of the hexagonal lattice and all its shortest edges. Middle: a partition into one and two thirds of the points, with MST-ratio converging to $(2+\sqrt{3})/3 = 1.245\ldots$. Right: a partition into one and three quarters of the points, with MST-ratio converging to $1.25$.
• Figure 2: The portion of the horizontally stretched hexagonal lattice, $\Lambda{\hbox{$\Lambda$}}$, and the subset of blue points, $B$, inside a square centered at the origin. The edges show the union of all possible minimum spanning trees of the blue points.
• Figure 3: Four partitions of the hexagonal lattice into two sets, in which we draw each (blue) point of the smaller set with its hexagonal neighborhood. The proportions of blue versus white points are $1:2$ in the upper middle, $1:3$ on the left, $1:6$ on the right, and $1:8$ in the lower middle. The corresponding MST-ratios are approximately $1.245$, $1.25$, $1.236$, and $1.222$, in this sequence.
• Figure 4: The hexagonal lattice of $36$ points on the torus, obtained by gluing opposite sides of the rhombus. The sublattice with twice the distance between neighboring points in shown in blue.
• Figure 5: The unit disk under the hexagonal distance in the plane. The edges that connect the origin to the corners at $\pm (\bf x{\hbox{$\bf x$}}-\bf y{\hbox{$\bf y$}})$, $\pm (\bf y{\hbox{$\bf y$}}-\bf z{\hbox{$\bf z$}})$, $\pm (\bf z{\hbox{$\bf z$}}-\bf x{\hbox{$\bf x$}})$ decompose the hexagon into six equilateral triangles, whose barycenters are $\pm \bf x{\hbox{$\bf x$}}$, $\pm \bf y{\hbox{$\bf y$}}$, $\pm\bf z{\hbox{$\bf z$}}$.

Theorems & Definitions (27)

• Definition
• Theorem 1
• proof
• Definition
• Theorem 2
• Claim 1
• proof
• Lemma 1
• proof
• Claim 2