On the L(h,k)-Labeling Co-Comparability Graphs and Circular-Arc Graphs

Given two nonnegative integers h and k, an L(h, k)-labeling of a graph G = (V, E) is a map from V to a set of integer 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 thought, in this article we deal with co-comparability graphs, its subclass of interval graphs, and circular-arc graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k)-labeling of co-comparability and circular-arc graphs. In particular, we provide the first algorithm to L(h, k)-label co-comparability, interval, and circular-arc graphs with a bounded number of colors. Finally, in the special case where k = 1 and G is an interval graph, our algorithm improves on the best previously-known ones using a number of colors that is at most twice the optimum