L(p,q) labeling of d-Dimensional Grids
Résumé
In this paper, we address the problem of lambda 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 dier at least by a value p, while stations that are within a larger radius r0 > r must use frequencies that dier 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 diering by at least q, while minimizing the value of M. M is then called the lambda number of G, and is denoted by lambda_p^q (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 lambda number of an L(p; q) labeling for any d \geq 1 and p; q \geq 0. Some of these results are optimal (namely, in the following cases : (1) p = 0, (2) q = 0, (3) q = 1 (4) p; q\geq 1, p = alpha q with 1 \leq alpha \leq 2d and (5) p \geq 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. [DMP+02], and generalizes results from [BPT00] and [DMP+02]. We also apply our results to get upper bounds for the L(p; q) labeling of d-dimensional hypercubes.
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