Exact Solution of a Class of Frequency Assignment Problems in Cellular Networks

For any non negative real values h and k, an L(h, k)-labeling of a graph G = (V,E) is a function L : V → IR 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)-labeling is the difference between the largest and the smallest value of L, so it is not restrictive to assume 0 as the smallest value of L. We denote by λh,k(G) the smallest real λ such that graph G has an L(h, k)-labeling of span λ. The aim of the L(h, k)-problem is to satisfy the distance constraints using the minimum span. In this paper, we study L(h, k)-labeling problem on regular grids of degree 3, 4, 6, and 8 solving several open problems left in the literature.