Optimal L(d,1)-labelings of certain direct products of cycles and Cartesian products of cycles

An L(d,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least d and those at a distance of two receive labels that differ by at least one, where d⩾1. Let λ1d(G) denote the least λ such that G admits an L(d,1)-labeling using labels from {0,1,…,λ}. We prove that (i) if d⩾1, k⩾2 and m0,…,mk-1 are each a multiple of 2k+2d-1, then λ1d(Cm0×⋯×Cmk-1)⩽2k+2d-2, with equality if 1⩽d⩽2k, and (ii) if d⩾1, k⩾1 and m0,…,mk-1 are each a multiple of 2k+2d-1, then λ1d(Cm0□⋯□Cmk-1)⩽2k+2d-2, with equality if 1⩽d⩽2k.