L(p, q) labeling of d-dimensional grids

In this paper, we address the problem of λ labelings, that was introduced in the context of frequency assignment for telecommunication networks. In this model, stations within a given radius r must use frequencies that differ at least by a value p, while stations that are within a larger radius r′ > r must use frequencies that differ by at least another value q. The aim is to minimize the span of frequencies used in the network. This can be modeled by a graph coloring problem, called the L(p, q) labeling, where one wants to label vertices of the graph G modeling the network by integers in the range [0; M ], in such a way that (1) neighbors in G are assigned colors differing by at least p and (2) vertices at distance 2 in G are assigned colors differing by at least q, while minimizing the value of M . M is then called the λ number of G, and is denoted by λpq(G). In this paper, we study the L(p, q) labeling for a specific class of networks, namely the d-dimensional grid Gd = G[n1, n2 . . . nd]. We give bounds on the value of the λ number of an L(p, q) labeling for any d ≥ 1 and p, q ≥ 0. Some of these results are optimal (namely, in the following cases : (1) p = 0, (2) q = 0, (3) q = 1 (4) p, q ≥ 1, p = α · q with 1 ≤ α ≤ 2d and (5) p ≥ 2dq + 1) ; when the results we obtain are not optimal, we observe that the bounds differ by an additive factor never exceeding 2q − 2. The optimal result we obtain in the case q = 1 answers an open problem stated by Dubhashi et al. [DMP02], and generalizes results from [BPT00] and [DMP02]. We also apply our results to get upper bounds for the L(p, q) labeling of d-dimensional hypercubes.