### Citations

3233 | The capacity of wireless networks - Gupta, Kumar - 2000 |

1989 |
Reducibility Among Combinatorial Problems
- Karp
- 1972
(Show Context)
Citation Context ...iables minus sum of negative variables must be not less than one (e.g. (a ∨ b ∨ ¬c) becomes a + b + (1− c) ≥ 1). Indeed, binary programming is among the original 21 NPcomplete problems put by Karp in =-=[Kar72]-=-. Still, due to wide application over practical problems, there is a big interest in solving these models. Many exact methods have been proposed: cutting plane, branch and bound, column generation and... |

431 |
A sweepline algorithm for voronoi diagrams
- Fortune
- 1987
(Show Context)
Citation Context ...ints uniformly distributed over a square. These represent the vertices of the graph. To obtain the edges, compute a Delaunay triangulation. This can be done e.g. with Fortune’s algorithm described in =-=[For87]-=- in O(n log n) time. See Figure A.9a for a depiction of a fragment of such graph. Vertices are arranged according to the positions of original random points. Dotted lines delimit corresponding Voronoi... |

420 | New methods to color the vertices of a graph - Brelaz - 1979 |

258 |
On colouring the nodes of a network
- Brooks
- 1941
(Show Context)
Citation Context ...1s- 1 7sNo vs20 13 giving ∆(G,w) ≥ ∆(G,w) + gcd(w), a contradiction. The result follows. Note that when all weights are unit, we obtain the bound for the improper colouring derived in [Lov66]. Brooks =-=[Bro41]-=- proved that for a connected graph G, χ(G) = ∆(G) + 1 if, and only if, G is complete or an odd cycle. One could wonder for which edge-weighted graphs the bound we provided in Theorem 5 is tight. Howev... |

238 | Stochastic geometry and random graphs for the analysis and design of wireless networks - Haenggi, Andrews, et al. - 2009 |

108 | Models and solution techniques for the frequency assignment problem. 4OR, 1(4):261– 317, - Aardal, Hoesel, et al. - 2003 |

97 |
On decompositions of graphs
- Lovász
- 1966
(Show Context)
Citation Context ...asily some upper bounds on the weighted t-improper chromatic number and the minimum k-threshold of an edge-weighted graph (G,w). It is a folklore result that χ(G) ≤ ∆(G) + 1, for any graph G. Lovász =-=[Lov66]-=- extended this result for Improper Colouring problem using w-balanced colouring. He proved that χl(G) ≤ d∆(G)+1l+1 e. In what follows, we extend this result to weighted improper colouring. Theorem 5. ... |

90 |
A survey on labeling graphs with a condition at distance two
- Yeh
- 2006
(Show Context)
Citation Context ...ssigned to them must be greater than some function which usually depends on their distance. There is a strong relationship between most of these variations and the L(p1, . . . , pd)-labelling problem =-=[Yeh06]-=-. In this problem, the goal is to find a colouring of the vertices of a given graph G, in such a way that the difference between the colours assigned to vertices at distance i is at least pi, for ever... |

82 | Stochastic geometry and architecture of communication networks - Baccelli, Klein, et al. - 1997 |

42 | Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency - Cowen, Cowen, et al. - 1986 |

29 | Defective coloring revisited - Cowen, Goddard, et al. - 1995 |

14 | Frequency assignment in mobile radio systems using branch-and-cut techniques - Fischetti, Lepschy, et al. - 2000 |

12 | L(2,1)-labelling of graphs
- Havet, Reed, et al.
- 2008
(Show Context)
Citation Context ...all equal to 1 or M . Let GM be the subgraph of G induced by the edges of weight M ; is it true that if ∆(GM) << ∆(G), then χt(G,w) ≤ χt(G) ≤ ⌈ ∆(G,w)+1 t+1 ⌉ ? A similar result for L(p, 1)-labelling =-=[HRS08]-=- suggests it could be true. Note that the problem can also be solved algorithmically for other classes of graphs and for other functions of interference. We started looking in this direction in [ABG+1... |

12 | An enumerative algorithm for the frequency assignment problem
- Mannino, Sassano
- 2003
(Show Context)
Citation Context ...ghted Improper Colouring, because we consider the co-channel interference, i.e. penalties, just between each vertex and its neighbourhood. The two closest related works we found in the literature are =-=[MS03]-=- and [FLM+00]. However, they both apply penalties over co-channel interference, but also to the adjacent channel interference, i.e. when the colours of adjacent vertices differ by one unit. Moreover, ... |

10 | Improper colorings of graphs. In - Woodall - 1990 |

9 | Quasioptimal bandwidth allocation for multi-spot MFTDMA satellites - Alouf, Altman, et al. - 2005 |

9 | About a Brooks-type theorem for improper colouring
- Correa, Havet, et al.
- 2009
(Show Context)
Citation Context ... connected graph G, χ(G) = ∆(G) + 1 if, and only if, G is complete or an odd cycle. One could wonder for which edge-weighted graphs the bound we provided in Theorem 5 is tight. However, Correa et al. =-=[CHS09]-=- already showed that it is NP-complete to determine if the improper chromatic number of a graph G attains the upper bound of Lovász, which is a particular case of Weighted Improper colouring, i.e. of... |

3 | Frédéric Giroire, Frédéric Havet, Dorian Mazauric, and Remigiusz Modrzejewski. Weighted improper colouring - Araujo, Bermond - 2011 |