Abstract
The edge-isoperimetric problem has long been solved for cartesian powers of the cycles C 3 and C 4, for which the lexicographic order is the optimal order, and powers of the cycles C n with n>5, which do not have nested optimal subsets. For powers of C 5, it is clear that the lexicographic order is not optimal. We present a solution to the edge-isoperimetric problem for powers of C 5 in the form of an optimal order for the vertices. We then prove that discrete tori of the forms C 5 i × C 4 j × C 3 k and C n × C 5 i × C 4 j × C 3 k have nested optimal subsets for n>5, i,j,k⩾0 , and give an optimal order for members of that class. We conjecture that these are the only discrete tori which have nested optimal subsets.
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