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

Abstract

Given any fixed nonnegative integer values h and k, the L(h, k)labelling problem consists in an assignment of nonnegative 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 minimum span. The L(h, k)-labelling problem has been intensively 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 previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

Distance Two Labeling on the Square of a Cycle

“…Firstly, we establish the lower and upper bound. [8,10,12,14,18,16,13,11,9,7,3,5], A 6 = [8,10,12,14,17,19,15,13,11,9,7,3,5]. Now we give an L(2, 1)-labeling of C 2 n with edge span 6, as shown in Table 1.…”

“…Therefore, the problem has been studied for many special classes of graphs, such as regular grids [1,2], product graphs [10,14], trees [4,18], planar graphs [17], generalized flowers [11], permutation and bipartite permutation graphs [15] and so on. For more details, one may refer to the surveys [3,19].…”

Algorithmic aspects of distance constrained labeling: a survey

“…In this context, L(2, 1)-labeling is generalized into L(p, q)-labeling for arbitrary nonnegative integers p and q, and in fact, we can see that L(1, 0)-labeling (L(p, 0)-labeling, actually) is equivalent to the classical vertex coloring. We can find a lot of related results on L(p, q)-labelings in comprehensive surveys by Calamoneri [11,12] and by Yeh [68]. The survey paper [12] is still updated and we can download the latest version from a web page 1 .…”

“…We can find a lot of related results on L(p, q)-labelings in comprehensive surveys by Calamoneri [11,12] and by Yeh [68]. The survey paper [12] is still updated and we can download the latest version from a web page 1 . The current latest version is ver.…”

Computational complexity of distance edge labeling

“…The second neighborhood is the set of vertices (or edges) at distance at most 2. For a survey about distance labelings, we refer to the article by Tiziana Calamoneri [2], as well as her online survey [3].…”