Optimal L(h, k)-Labeling of Regular Grids

The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that ’close’ nodes have labels which differ by at least k, and ’very close’ nodes have labels which differ by at least h. The span of an L(h, k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h, k)labeling problem has been intensively studied in some special cases, i.e. when k = 0 (vertex coloring), h = k (vertex coloring the square of the graph) and h = 2k (radioor λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h, k)-labeling problem on cellular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds. For the sake of completeness, we study also hexagonal and squared grids.