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15
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 ..."
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Cited by 29 (3 self)
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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.
On the L(p, 1)labelling of graphs
 IN PROC. EUR. CONF. ON COMBINATORICS, GRAPH THEORY AND APPLICATIONS (EUROCOMB ’05
, 2005
"... In this paper we improve the best known bound for the L(p, 1)labelling of graphs with given maximal degree. ..."
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Cited by 24 (0 self)
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In this paper we improve the best known bound for the L(p, 1)labelling of graphs with given maximal degree.
Coloring squares of planar graphs with girth six
 SUBMITTED TO EUROPEAN JOURNAL OF COMBINATORICS
"... Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off ..."
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Cited by 19 (4 self)
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Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree ∆ ≥ M(g) is (∆+1)colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ {5, 6}. We show that the conjecture for g = 6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is ( ∆ + 2)colorable.
Graph Labellings with Variable Weights, a Survey
, 2007
"... Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings ..."
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Cited by 12 (1 self)
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Graph labellings form an important graph theory model for the channel assignment problem. An optimum labelling usually depends on one or more parameters that ensure minimum separations between frequencies assigned to nearby transmitters. The study of spans and of the structure of optimum labellings as functions of such parameters has attracted substantial attention from researchers, leading to the introduction of real number graph labellings and λgraphs. We survey recent results obtained in this area. The concept of real number graph labellings was introduced a few years ago, and in the sequel, a more general concept of λgraphs appeared. Though the two concepts are quite new, they are so natural that there are already many results on each. In fact, even some older results fall in this area, but their authors used a different mathematical language to state their achievements. Since many of these results are so recent that they are just appearing in various journals, we would like to offer the reader a single reference for the state of art as well as to draw attention to some older results that fall in this area.
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 ..."
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Cited by 8 (3 self)
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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.
Optimal real number graph labelings of a subfamily of Kneser graphs
, 2006
"... A notion of real number graph labelings captures the dependence of the span of an optimal channel assignment on the separations that are required between frequencies assigned to close transmitters. We determine the spans of such optimal labelings for a subfamily of Kneser graphs formed by the comple ..."
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Cited by 1 (0 self)
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A notion of real number graph labelings captures the dependence of the span of an optimal channel assignment on the separations that are required between frequencies assigned to close transmitters. We determine the spans of such optimal labelings for a subfamily of Kneser graphs formed by the complements of the line graphs of complete graphs. This subfamily contains (among others) the Petersen graph.
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 ..."
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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.
On Colorings of graph fractional powers
, 2009
"... For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic nu ..."
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For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n) = ω(G m n) provided that G is a connected graph with < 1. It is also shown that this conjecture is true in some special cases. ∆(G) ≥ 3 and m