Description
Abstract: We present two results related to an edge-isoperimetric question for Cayley graphs on d-dimensional lattices asked by Barber and Erde (2018). For any 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 the integer lattice, there is always an ordering of the points of the lattice such that for each n, the first n points in the sequence minimize the edge boundary among all subsets of size n. Our first result is a simple negative answer in each dimension at least 2. Our second result is a relatively complicated positive example in dimension 2. This is joint work with Cameron Strachan (LSE).
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