Lattice spanners of low degree

TL;DR

This paper investigates the degree 3 and 4 dilations of infinite square and hexagonal lattices, providing bounds and exact values for their lattice spanners, with a focus on planar tilings constrained to these degrees.

Contribution

It establishes new bounds and exact values for the degree 3 and 4 dilations of lattice spanners in square and hexagonal lattices, using planar tilings.

Findings

For the square lattice, 1+√2 ≤ δ₀(Λ,3) ≤ 2.6065 and δ₀(Λ,4)=√2.

For the hexagonal lattice, δ₀(Λ,3)=1+√3 and δ₀(Λ,4)=2.

Abstract

Let $\delta_0(P,k)$ denote the degree $k$ dilation of a point set $P$ in the domain of plane geometric spanners. If $\Lambda$ is the infinite square lattice, it is shown that $1+\sqrt{2} \leq \delta_0(\Lambda,3) \leq (3+2\sqrt2) \, 5^{-1/2} = 2.6065\ldots$ and $\delta_0(\Lambda,4) = \sqrt{2}$. If $\Lambda$ is the infinite hexagonal lattice, it is shown that $\delta_0(\Lambda,3) = 1+\sqrt{3}$ and $\delta_0(\Lambda,4) = 2$. All our constructions are planar lattice tilings constrained to degree $3$ or $4$.