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 d1 (G) denote the least such that G admits an L(d,1)-labeling using labels from {0, 1, . . . , }. We prove that (i) if d 1, k 2 and m0, . . . , mk−1 are each a multiple of 2k + 2d − 1, then d1 (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 d1 (Cm0 · · · Cmk−1) 2k + 2d − 2, with equality if 1 d 2k. © 2005 Elsevier B.V. All rights reserved.