A note on optimal degree-three spanners of the square lattice

TL;DR

This paper proves that the degree-three dilation of the square lattice is 1+√2, disproving a previous conjecture, and provides a computer-assisted proof of a local-global property for optimal dilation graphs.

Contribution

It establishes the exact degree-three dilation for the square lattice and introduces a computer-assisted method to analyze geometric graphs with optimal dilation.

Findings

Degree-three dilation of the square lattice is 1+√2

Disproves Dumitrescu and Ghosh's conjecture

Provides a computer-assisted proof of a local-global property

Abstract

In this short note, we prove that the degree-three dilation of the square lattice $\mathbb{Z}^2$ is $1+\sqrt{2}$. This disproves a conjecture of Dumitrescu and Ghosh. We give a computer-assisted proof of a local-global property for the uncountable set of geometric graphs achieving the optimal dilation.