Edge Isoperimetric Inequalities for Powers of the Hypercube

For positive integers n and r, we let Q(n)(r) denote the rth power of the n-dimensional discrete hypercube graph, i.e., the graph with vertex-set {0,1}(n), where two 0-1 vectors are joined if they are Hamming distance at most r apart. We study edge isoperimetric inequalities for this graph. Harper, Bernstein, Lindsey and Hart proved a best-possible edge isoperimetric inequality for this graph in the case r = 1. For each r >= 2, we obtain an edge isoperimetric inequality for Q(n)(r); our inequality is tight up to a constant factor depending only upon r. Our techniques also yield an edge isoperimetric inequality for the 'Kleitman-West graph' (the graph whose vertices are all the k-element subsets of {1, 2, ..., n}, where two k-element sets have an edge between them if they have symmetric difference of size two); this inequality is sharp up to a factor of 2 + o(1) for sets of size ((n-s)(k-s)), where k = o(n) and s is an element of N.