The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography

Abstract

Given any fixed non-negative integer values h and k, the L(h, k)-labelling problem consists in an assignment of non-negative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2-length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with a minimum span. The L(h, k)-labelling problem has intensively been studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previously published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.