We present two results related to an edge isoperimetric question for Cayley graphs on the integer lattice asked by Barber and Erde (Discrete Anal Paper no. 7:16, 2018). For any (undirected) graph G, the edge boundary of a subset of vertices S is the number of edges between S and its complement in G. Barber and Erde asked whether for any Cayley graph on \(\mathbb {Z}^d\) , there is always an ordering of \(\mathbb {Z}^d\) such that for each n, the first n terms minimize the edge boundary among all subsets of size n. First, we present an example of a Cayley graph \(G_d\) on \(\mathbb {Z}^d\) (for all \(d\ge 2\) ) for which there is no such ordering. Furthermore, we show that for all n and any optimal n-vertex subset \(S_n\) of \(G_d\) , there is no infinite sequence \(S_n\subset S_{n+1}\subset S_{n+2}\subset \cdots \) of optimal sets \(S_i\) , where \(|S_i|=i\) for \(i\ge n\) . This is to be contrasted with the positive result in \(\mathbb {Z}^1\) shown by Joseph Briggs and Chris Wells [arXiv:2402.14087]. Our second result is a positive example for the unit-length triangular lattice (which is isomorphic to \(\mathbb {Z}^2\) ) where two vertices are connected by an edge if their distance is 1 or \(\sqrt{3}\) . We show that this graph has such an ordering. This is the most complicated example known to us of a two-dimensional Cayley graph for which an ordering exists.
