Isoperimetry in integer lattices

Ben Barber, Joshua Erde

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
295 Downloads (Pure)


The edge isoperimetric problem for a graph G is to determine, for each n, the minimum number of edges leaving any set of n vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example when G is the usual integer lattice. We solve the edge isoperimetric problem asymptotically for every Cayley graph on Zd. The near-optimal shapes that we exhibit are zonotopes generated by line segments corresponding to the generators of the Cayley graph.
Original languageEnglish
Article number3555
Number of pages16
JournalDiscrete Analysis
Publication statusPublished - 20 Apr 2018


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