The edge-isoperimetric problem for discrete tori Original Research Article

The edge-isoperimetric problem has long been solved for cartesian powers of the cycles C3 and C4, for which the lexicographic order is the optimal order, and powers of the cycles Cn with n>5, which do not have nested optimal subsets. For powers of C5, it is clear that the lexicographic order is not optimal. We present a solution to the edge-isoperimetric problem for powers of C5 in the form of an optimal order for the vertices. We then prove that discrete tori of the forms C5i × C4j × C3k and Cn × C5i × C4j × C3k 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.