New bounds for the L(h, k) number of regular grids
by Tiziana Calamoneri, Saverio Caminiti, Guillaume Fertin
International Journal of Mobile Network Design and Innovation (IJMNDI), Vol. 1, No. 2, 2006

Abstract: For any non-negative real values h and k, an L(h, k)-labelling of a graph G = (V ,E) is a function L : V → R such that |L(u) − L(v)| ≥ h if (u, v) ∈ E and |L(u) − L(v)| ≥ k if there exists w ∈ V such that (u,w) ∈ E and (w, v) ∈ E. The span of an L(h, k)-labelling is the difference between the largest and the smallest value of L. We denote by λ_h,k(G) the smallest real λ such that graph G has an L(h, k)-labelling of span λ. The aim of the L(h, k)-labelling problem is to satisfy the distance constraints using the minimum span. In this paper, we study the L (h, k)-labelling problem on regular grids of degree 3, 4 and 6 for those values of h and k whose λ_h,k is either not known or not tight. We also initiate the study of the problem for grids of degree 8. For all considered grids, in some cases we provide exact results, while in the other ones we give very close upper and lower bounds.

Online publication date: Tue, 05-Sep-2006

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