Characterizing imprimitive partition designs of binary Hamming graphs

Abstract

Let G=( V, E) be a binary Hamming graph (or the 1-skeleton of a hypercube). A partition design of G with adjacency matrix M=( m ij ) 1≤ i, j≤ r is defined as a partition { Y 1,…, Y r } of the vertex set V such that for every x∈ Y i we have that |{ y∈ Y j ∣( x, y)∈ E}|= m ij ; this holds for 1≤ i, j≤ r. Let Y be a partition design with adjacency matrix M. For every t≥2 we construct a partition design Y t with adjacency matrix tM, and we describe when Y t is the unique partition design with adjacency matrix tM.