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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.
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.
Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph
, 2006
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L(h,1,1)Labeling of Outerplanar Graphs
"... An L(h,1, 1)labeling of a graph is an assignment of labels from the set of integers {0, · · · , λ} to the vertices of the graph such that adjacent vertices are assigned integers of at least distance h ≥ 1 apart and all vertices of distance three or less must be assigned different labels. The ai ..."
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An L(h,1, 1)labeling of a graph is an assignment of labels from the set of integers {0, · · · , λ} to the vertices of the graph such that adjacent vertices are assigned integers of at least distance h ≥ 1 apart and all vertices of distance three or less must be assigned different labels. The aim of the L(h, 1,1)labeling problem is to minimize λ, denoted by λh,1,1 and called span of the L(h,1, 1)labeling. As outerplanar graphs have bounded treewidth, the L(1, 1,1)labeling problem on outerplanar graphs can be exactly solved in O(n 3), but the multiplicative factor depends on the maximum degree ∆ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h,1, 1)labeling for outerplanar graphs that is within additive constants of the optimum values.
Vertex coloring the square of outerplanar graphs of low degree
, 2010
"... Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree ∆ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar g ..."
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Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree ∆ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree ∆ != 6 is known. Hence, this mentioned chromatic number of 7 is the last and only unknown upper bound of the chromatic number in terms of ∆.
Radiocolorings in Periodic Planar Graphs: PSPACECompleteness and Efficient Approximations for the Optimal Range of Frequencies
"... The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph ..."
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The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. A Radiocoloring (RC) of a graph G(V,E) is an assignment function Λ: V → IN such that Λ(u) − Λ(v)  ≥2, when u, v are neighbors in G, and Λ(u) − Λ(v)  ≥1 when the distance of u, v in G is two. The discrete number of frequencies used is called order and the range of frequencies used, span. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span (min span RCP) or the order (min order RCP). In this paper, we deal with an interesting, yet not examined until now, variation of the radiocoloring problem: that of satisfying frequency assignment requests which exhibit some periodic behavior. In this case, the interference graph (modelling interference between transmitters) is some (infinite) periodic graph. Infinite periodic graphs usually model finite networks that accept periodic (in time, e.g. daily) requests
A Note on Algebraic Hypercube Colorings
"... An L(1, 1)coloring of the ndimensional hypercube Qn assigns nodes of Qn which are at distance ≤ 2 with different colors. Such colorings find application, e.g., in frequency assignment in wireless networks and data distribution in parallel memory systems. Let χ 2 (Qn) be the minimum number of color ..."
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An L(1, 1)coloring of the ndimensional hypercube Qn assigns nodes of Qn which are at distance ≤ 2 with different colors. Such colorings find application, e.g., in frequency assignment in wireless networks and data distribution in parallel memory systems. Let χ 2 (Qn) be the minimum number of colors used in any L(1, 1)coloring. Finding the exact value of χ 2 (Qn) is still an open problem, and only 2approximate solutions are currently known. In this paper we expose some connections between group theory and the L(1, 1)coloring problem. Namely, we unfold the algebraic structure on which the best available L(1, 1)coloring algorithms of Qn are based, thus giving a group theoretic flavour to existing L(1, 1)colorings. We show that identifying groups such that the inverse of each element is the element itself yields a simple and efficient way to obtain L(1, 1)colorings of the hypercube. We also prove that such colorings are balanced and that every coloring algorithm based on this algebraic structure cannot improve the current upper bound on χ 2 (Qn), independently of the choice of the group operation.
A Note on Strongly Simplicial Vertices of Powers of Trees
"... For a tree T and an integer k ≥ 1, it is well known that the kth power T k of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of T k, thereby characterizing all strong elimination o ..."
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For a tree T and an integer k ≥ 1, it is well known that the kth power T k of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of T k, thereby characterizing all strong elimination orderings of the vertices of T k.
Tree tspanners in Outerplanar Graphs via Supply Demand Partition
"... Abstract. A tree tspanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree tspanner problem is to compute a tree tspanner of G if one e ..."
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Abstract. A tree tspanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree tspanner problem is to compute a tree tspanner of G if one exists. This decision problem is known to be NPcomplete even in the restricted class of unweighted planar graphs. We present a lineartime reduction from tree tspanner in outerplanar graphs to the supplydemand tree partition problem. Based on this reduction, we obtain a lineartime algorithm to solve tree tspanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree tspanner can be found in O(n logn) time. 1