Edge Boundaries for a Family of Graphs on $\mathbb{Z}^n$

TL;DR

This paper investigates the edge boundary of finite sets in a specific lattice graph on 5^n, showing that optimal boundary sets can be chosen without gaps in coordinate directions, advancing isoperimetric understanding.

Contribution

It provides an explicit calculation of edge boundaries for finite sets in 5^n and demonstrates the existence of optimal boundary sets without directional gaps.

Findings

Calculated edge boundary for finite sets in 5^n.

Proved the existence of gap-free optimal boundary sets in coordinate directions.

Abstract

We consider the family of graphs whose vertex set is $\mathbb{Z}^n$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1. Towards an edge isoperimetric inequality for this graph, we calculate the edge boundary of any finite set $S \subset \mathbb{Z}^n$. This boundary calculation leads to a desire to show that a set with optimal edge boundary has no ``gaps'' in any direction $\epsilon \in \{-1,0,1\}^n, \epsilon \not=0$. We show that one can find a set with optimal edge boundary that does not have gaps in any direction $e_i$ (or $-e_i$) where $e_i$ is the standard basis vector.