Results 1  10
of
24
The L(h, k)Labelling Problem: A Survey and Annotated Bibliography
, 2006
"... 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 l ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
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.
Exact algorithms for L(2, 1)labeling of graphs
"... The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G =(V,E) into an interval of integers [0..k] isanL(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers that a ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G =(V,E) into an interval of integers [0..k] isanL(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k ≥ 4, deciding the existence of such a labeling is an NPcomplete problem. We present exact exponential time algorithms that are faster than the naive O((k +1) n) algorithm that would try all possible mappings. The improvement is best seen in the first NPcomplete case of k = 4 – here the running time of our algorithm is O(1.3161^n).
Colourings of the Cartesian product of graphs and multiplicative Sidon sets
, 2005
"... Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choic ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choices of F, several wellknown types of colourings fit into this framework, including acyclic colourings, star colourings, and distance2 colourings. This paper studies Ffree colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1, G2,..., Gd. Our main result establishes an upper bound on the Ffree chromatic number of H in terms of the maximum Ffree chromatic number of the Gi and the following numbertheoretic concept. A set S of natural numbers is kmultiplicative Sidon if ax = by implies a = b and x = y whenever x,y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a kmultiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k·max S. We then prove that the maximum density of a kmultiplicative Sidon set is Θ(1/log k). It follows that χ(H, F) ≤ O(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.
Intersection Graphs of Pseudosegments: Chordal Graphs
, 2010
"... We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graphs of subpaths on a tree are pseudosegment intersection graphs. We then study the li ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graphs of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We identify certain intersection graphs of substars ofastar whichare notrepresentable as intersection graphs of pseudosegments. The degree of the substars in these examples, however, has to be large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graphs of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different ksegments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n²).
Improved Bounds on the L(2, 1)Number of Direct and Strong Products of Graphs
, 2007
"... The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex label ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An L(2, 1)labeling of a graph G is a function f from the vertex set V (G) to the set of all nonnegative integers such that f(x) − f(y)  ≥ 2 if d(x, y) = 1 and f(x) − f(y)  ≥ 1 if d(x, y) = 2, where d(x, y) denotes the distance between x and y in G. The L(2, 1)labeling number λ(G) of G is the smallest number k such that G has an L(2, 1)labeling with max{f(v) : v ∈ V (G)} = k. This paper considers the graph formed by the direct product and the strong product of two graphs and gets better bounds than those of [14] with refined approaches.
Exact Algorithms For L(2, 1)Labeling of Graphs
, 2007
"... The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G = (V, E) into an interval of integers{ 0,..., k} is an L(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G = (V, E) into an interval of integers{ 0,..., k} is an L(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k> = 4, deciding the existence of such a labeling is an NPcomplete problem. We present exact exponential time algorithms that are faster than the naive O((k + 1)n) algorithm that would try all possible mappings. The improvement is best seen in the first NPcomplete case of k = 4 here the running time of our algorithm is O(1.3006n). Furthermore we show that dynamic programming can be used to establish an O(3.8730n) algorithm to compute an optimal L(2, 1)labeling.
On L(0,1)Labelling of Cactus Graphs
 Communications and Network
, 2012
"... Anlabelling of a graph is an assignment of nonnegative integers to the vertices of such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The sp ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Anlabelling of a graph is an assignment of nonnegative integers to the vertices of such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an 0,1L G G 0,1Llabelling is the maximum label number assigned to any vertex of. The G 0,1L G 0,1,G 1 k 1 labelling number of a graph, denoted by is the least integer such that G has anlabelling of span. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph bylabelling and have shown that, k 0,L 0,L 0,1 G1 for a cactus graph, where is the degree of the graph. 0,1L =,G V E G
Distance Constrained Labelings of K4minor Free Graphs
, 2006
"... Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Motivated by previous results on distance constrained labelings and coloring of squares of K4minor free graphs, we show that for every p> = q> = 1, there exists \Delta 0 such that every K4minor free graph G with maximum degree \Delta> = \Delta 0 has an L(p, q)labeling of span at most qb3\Delta (G)/2c. The obtained bound is the best possible.