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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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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 λ-graphs. 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 λ-graphs 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.
Bounds for the real number graph labellings and application to labellings of the triangular lattice
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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.
Distance-1 Constrained Channel Assignment in Single Radio Wireless Mesh Networks
"... Abstract — This paper addresses channel assignment and random medium access design for single-radio multi-channel mesh networks. Two prior approaches include: (i) designing MAC protocols that dynamically select channels based on local information and (ii) partitioning the mesh into subnetworks with ..."
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Abstract — This paper addresses channel assignment and random medium access design for single-radio multi-channel mesh networks. Two prior approaches include: (i) designing MAC protocols that dynamically select channels based on local information and (ii) partitioning the mesh into subnetworks with different channels and using 802.11 as the medium access protocol. Both of these approaches suffer from limited throughput improvement; the first approach due to wrong or incomplete channel state information that inherently arises in a multi-hop wireless environment, while the second approach due to high interference within each subnetwork. In this paper, we first introduce D1C-CA, Distance-1 Constrained Channel Assignment. D1C-CA statically assigns channels to a set of links as a function of physical connectivity, contention, and the unique gateway functionality of mesh networks, i.e, all internet (non-local) traffic has a gateway node as its source or destination. To design D1C-CA, we model the channel assignment problem as a new form of graph edge coloring in which edges at distance one are constrained. We prove that the problem is NP-complete and design an efficient heuristic solution for mesh networks. Second, we design an asynchronous control-channel-based MAC protocol that solves multi-channel coordination problems and employs the proposed channel assignment algorithm. Finally, we investigate the performance of our approach through extensive simulations and show considerable performance improvements compared to alternate schemes. I.
Complexity of Generalized Colourings of Chordal Graphs
, 2008
"... The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P1,...,Pk (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subg ..."
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The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P1,...,Pk (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subgraph of G on Vi belongs to Pi. It can be seen that GCOL generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The structure of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets Pi are characterized by a single forbidden induced subgraph. We show, that for k = 2, all such problems where the forbidden graphs have at most three vertices are polynomial time solvable in chordal graphs, whereas, it is known that almost all of them are NP-complete in general. On the other hand, we show infinite families of such problems which are NP-complete in chordal graphs. By combining a polynomial algorithm and an
A distance labelling problem for hypercubes
"... Let i1 ≥ i2 ≥ i3 ≥ 1 be integers. An L(i1, i2, i3)-labelling of a graph G = (V, E) is a mapping φ: V → {0, 1, 2,...} such that |φ(u) − φ(v) | ≥ it for any u, v ∈ V with d(u, v) = t, t = 1, 2, 3, where d(u, v) is the distance in G between u and v. The integer φ(v) is called the label assigned to v ..."
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Let i1 ≥ i2 ≥ i3 ≥ 1 be integers. An L(i1, i2, i3)-labelling of a graph G = (V, E) is a mapping φ: V → {0, 1, 2,...} such that |φ(u) − φ(v) | ≥ it for any u, v ∈ V with d(u, v) = t, t = 1, 2, 3, where d(u, v) is the distance in G between u and v. The integer φ(v) is called the label assigned to v under φ, and the difference between the largest and the smallest labels is called the span of φ. The problem of finding the minimum span, λi1,i2,i3(G), over all L(i1, i2, i3)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i1, i2, i3)labelling was originated from recent studies of the scalability of optical networks. In this paper we study the L(i1, i2, i3)-labelling problem for hypercubes Qd (d ≥ 3) and obtain upper and lower bounds on λi1,i2,i3(Qd) for any (i1, i2, i3).
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 ..."
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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.
Distance labelling problems for hypercubes and Hamming graphs-a survey
- Elec. Notes Discrete Math
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