On the L(h, k)-labeling of co-comparability graphs

Given two non negative integers h and k, an L(h, k)-labeling of a graph G = (V, E) is a map from V to a set of labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k)-labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h, k)-labeling problem is NP-complete, it is important to investigate classes of graphs for which the problem can be solved efficiently. Along this line of though, in this paper we deal with co-comparability graphs and two of its subclasses: interval graphs and unit-interval graphs. Specifically, we provide, in a constructive way, the first upper bounds on the L(h, k)-number of co-comparability graphs and interval graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k)-labeling of co-comparability graphs. In the special case where k = 1, our result improves on the best previously-known approximation ratio for interval graphs.