Characterizing r-perfect codes in direct products of two and three cycles

An r-perfect code of a graph G = (V, E) is a set C ⊆ V such that the r-balls centered at vertices of C form a partition of V . It is proved that the direct product of Cm and Cn (r ≥ 1, m,n ≥ 2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r + 1)2 + r2 and that the direct product of Cm, Cn, and C` (r ≥ 1, m,n, ` ≥ 2r +1) contains an r-perfect code if and only if m, n, and ` are each a multiple of r3 + (r + 1)3. The corresponding r-codes are essentially unique. Also, r-perfect codes in C2r × Cn (r ≥ 2, n ≥ 2r) are characterized.