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17
List colouring squares of planar graphs
, 2008
"... In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs. ..."
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Cited by 28 (5 self)
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In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs.
L(2,1)labelling of graphs
 IN PROCEEDINGS OF THE ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHM (SODA 2008
, 2008
"... An L(2, 1)labelling of a graph is a function f from the vertex set to the positive integers such that f(x) − f(y)  ≥ 2 if dist(x, y) = 1 and f(x) − f(y)  ≥ 1 if dist(x, y) = 2, where dist(u, v) is the distance between the two vertices u and v in the graph G. The span of an L(2, 1)labelli ..."
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Cited by 13 (3 self)
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An L(2, 1)labelling of a graph is a function f from the vertex set to the positive integers such that f(x) − f(y)  ≥ 2 if dist(x, y) = 1 and f(x) − f(y)  ≥ 1 if dist(x, y) = 2, where dist(u, v) is the distance between the two vertices u and v in the graph G. The span of an L(2, 1)labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree ∆ ≥ 2 has an L(2, 1)labelling with span at most ∆2 + 1. We settle this conjecture for ∆ sufficiently large.
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.
Bounds for the real number graph labellings and application to labellings of the triangular lattice
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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 ..."
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Cited by 2 (0 self)
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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.
Real Number Channel Assignments for Lattices
, 2007
"... ... Portions of it have been obtained by other researchers for infinite regular lattices that model large planar networks. Here we present the complete function *(G; k, 1), for k> = 1 when G is the triangular, square, or hexagonal lattice. ..."
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Cited by 1 (0 self)
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... Portions of it have been obtained by other researchers for infinite regular lattices that model large planar networks. Here we present the complete function *(G; k, 1), for k> = 1 when G is the triangular, square, or hexagonal lattice.
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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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.
Determining the L(2, 1)span in polynomial space
 Discrete Appl. Math
, 2013
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Workshop on Graph colouring Problems Arising in Telecommunications: Final Report
, 2007
"... The channel assignment problem in radio or cellular phone networks is the following: we need to assign radio frequency bands to transmitters ( each station gets one channel which corresponds to an integer). In order to avoid interference, if two stations are very close, then the separation of the ch ..."
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The channel assignment problem in radio or cellular phone networks is the following: we need to assign radio frequency bands to transmitters ( each station gets one channel which corresponds to an integer). In order to avoid interference, if two stations are very close, then the separation of the channels assigned to them has to be large enough. Moreover, if two stations are close ( but not very close), then they must also receive channels that are sufficiently apart. Such problem may be modelled by L(p, q)labellings of a graph G. The vertices of this graph correspond to the transmitters and two vertices are linked by an edge if they are very close. Two vertices are then considered close if they are at distance 2 in the graph. Let dist(u, v) denote the distance between the two vertices u and v. An L(p, q)labelling of G is an integer assignment f to the vertex set V (G) such that: • f(u) − f(v)  ≥ p, if dist(u, v) = 1, and • f(u) − f(v)  ≥ q, if dist(u, v) = 2. As the separation between channels assigned to vertices at distance 2 cannot be smaller than the separation between channels assigned to vertices at distance 1, it is often assumed that p ≥ q. The span of f is the difference between the largest and the smallest labels of f plus one. The λp,qnumber