New bounds for the L(h, k) number of regular grids

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 λh,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 λh,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.