Regular grid subgraphs of maximal girth

TL;DR

This paper explores how to maximize the girth of regular subgraphs within high-dimensional integer lattices, providing bounds, examples, and extending to alternative lattice structures.

Contribution

It introduces new bounds and examples for maximal girth regular subgraphs of high-dimensional grids and considers alternative lattice structures.

Findings

Established bounds for girth in specific dimensions and degrees

Provided explicit examples of high-girth subgraphs

Extended analysis to alternative lattice types

Abstract

The unit-distance graph on the $n$-dimensional integer lattice $\mathbb{Z}^n$ is called the $n$-dimensional grid. We attempt to maximize the girth of a $k$-regular (possibly induced) subgraph of the $n$-dimensional grid, and provide examples and bounds for selected values of $n$ and $k$, along with more general results. A few cases involving alternative lattices are also considered.