Finding the chromatic number by means of critical graphs

Herrmann, Francine; Hertz, Alain

doi:10.1145/944618.944628

Cited by 28 publications

(13 citation statements)

References 7 publications

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“…As has been mentioned in [2] a greedy method would be to iteratively pick, in a graph , an uncolored vertex , and to color it with the smallest color which is not yet used by its neighbors . Such a coloring will obviously stay proper until the whole vertex set is colored, and it never uses more than ∆( ) + 1 different colors, where ∆( ) is the maximal degree of , as in the procedure no vertex will ever exclude more than ∆( ) colors (for more details see [3][4][5][6][7][8]). Consider is a graph.…”

Section: Introductionmentioning

confidence: 99%

New Algorithm for Chromatic Number of Graphs and their Applications

Rafat

2018

Journal of the Egyptian Mathematical Society

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Section: Introductionmentioning

confidence: 99%

New Algorithm for Chromatic Number of Graphs and their Applications

Rafat

2018

Journal of the Egyptian Mathematical Society

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“…When the problem of joint routing and network coding is extended to wireless ad-hoc network, interference between links must be controlled through scheduling, because of omnidirectional transmissions. Scheduling in wireless communications is modeled as a coloring problem in graph theory where two adjacent areas (sub graphs) cannot be painted with the same color [7]. Therefore the problem of routing and network coding should be solved simultaneously with scheduling [6].…”

Section: Introductionmentioning

confidence: 99%

Optimization of Scheduling and Routing in Wireless Ad-Hoc Networks Using Cubic Games

Karami

Glisic

2012

2012 IEEE Vehicular Technology Conference (VTC Fall)

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In this paper we present nonlinear 3 player game model for joint routing, network coding, and scheduling problem. To define such a game model, first routing and network coding are modeled by using a new approach based on compressed topology matrix that takes into account the inherent multicast gain of the network. Topology matrix includes the set of all possible paths, including network coded paths, from sources to their corresponding sinks. These paths are identified and compressed, and then by switching between some of them with appropriate usage rates (frequencies), achievable throughput is optimized.The scheduling is optimized by a new approach called network graph soft coloring. Soft graph coloring is designed by switching between different components of a wireless network graph, which we refer to as graph fractals, with appropriate usage rates. Therefore each link can be painted with more than one different colors selected with appropriate probabilities. In the proposed game which is a nonlinear cubic game, the strategy sets of the players are links, path, and network components. The outputs of this game model are mixed strategy vectors of the second and the third players at equilibrium. Strategy vector of the second player specifies optimum multi-path routing and network coding solution while mixed strategy vector of the third players indicates optimum switching rate among different network components or membership probabilities for optimal soft scheduling approach. Optimum throughput is the value of the proposed nonlinear cubic game at equilibrium. The proposed nonlinear cubic game is solved by extending fictitious playing method. Numerical and simulation results prove the superior performance of the proposed techniques compared to other conventional schemes.

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“…Hence in this regime one expects complexity to be monotonously increasing. It is generally believed that the complexity is maximal when k = χ(G) − 1 [8].…”

Section: Introduction and Previous Workmentioning

confidence: 99%

Graph coloring: The more colors, the better?

Szép

Mann

2010

2010 11th International Symposium on Computational Intelligence and Informatics (CINTI)

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In this paper, we investigate the algorithmic complexity of deciding colorability, as a function of the number of available colors. Intuitively, one may assume that the problem's complexity is highest around the chromatic number of the graph. We give substantial empirical evidence that this intuition is largely true, both for exact and heuristic graph coloring algorithms. We give a rigorous proof that the complexity of a class of exact algorithms is monotonously increasing in the number of available colors in the non-colorable case, and give a counter-example to demonstrate that the analogous claim does not always hold for colorable graphs.

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Finding the chromatic number by means of critical graphs

Cited by 28 publications

References 7 publications

New Algorithm for Chromatic Number of Graphs and their Applications

New Algorithm for Chromatic Number of Graphs and their Applications

Optimization of Scheduling and Routing in Wireless Ad-Hoc Networks Using Cubic Games

Graph coloring: The more colors, the better?

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