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

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 $ε\in \{-1,0,1\}^n, ε\not=0$. 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.

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