Polynomial Cases for the Vertex Coloring Problem

Karthick, T.; Maffray, Frédéric; Pastor, Lucas

doi:10.1007/s00453-018-0457-y

Cited by 19 publications

(14 citation statements)

References 35 publications

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“…Theorem 18 (Karthick, Maffray, and Pastor [18]). If G is a prime (P 5 , banner)-free graph of independence number at least 3, then G is perfect.…”

Section: Banner-free Graphsmentioning

confidence: 99%

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Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Brause¹,

Geißer²,

Schiermeyer³

2020

Preprint

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Given a set H of graphs, let f ⋆ H : N>0 → N>0 be the optimal χ-binding function of the class of H-free graphs, that is,In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal χ-bindung functions for subclasses of P5-free graphs and of (C5, C7, . . .)-free graphs. In particular, we prove the following for each ω ≥ 1:We also characterise, for each of our considered graph classes, all graphs G with χ(G) > χ(G − u) for each u ∈ V (G). From these structural results, we can prove Reed's conjecture -relating chromatic number, clique number, and maximum degree of a graph -for (P5, banner)-free graphs and for (P5, dart)-free graphs.

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“…Theorem 18 (Karthick, Maffray, and Pastor [18]). If G is a prime (P 5 , banner)-free graph of independence number at least 3, then G is perfect.…”

Section: Banner-free Graphsmentioning

confidence: 99%

“…Figure 5: Pastor [18], each such graph contains at most 18 vertices. However, in order to apply Lemma 11, we need a full characterisation of these graphs.…”

Section: Dart-free Graphsmentioning

confidence: 99%

Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Brause¹,

Geißer²,

Schiermeyer³

2020

Preprint

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“…Once the real-world problems are modeled with GCP, many exact [32]- [34] and heuristic approaches [35]- [38] can be used due to its NP-Hard complexity [6], [39], [40]. Although the proposed exact algorithms for solving graph coloring problem [41] can find the best solutions for small instances, they are expensive in terms of memory and time consumption for large instances [42].…”

Section: Related Workmentioning

confidence: 99%

Integrated Crossover Based Evolutionary Algorithm for Coloring Vertex-Weighted Graphs

Boz

Sungu

2020

IEEE Access

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Graph coloring is one of the main optimization problems widely studied in the literature. In this study, we propose a novel evolutionary algorithm called Integrated Crossover Based Evolutionary Algorithm with its unique crossover operator and local search technique for coloring vertex-weighted graphs. The integrated crossover operator targets to use the domain-specific information in the individuals and the local search technique aims to explore neighborhood solutions using weighted-swap operations. The performance of the proposed work is evaluated on synthetic benchmarks and DIMACS instances by comparing it with leading evolutionary algorithms from the literature. The experimental study indicates that our algorithm outperforms the related work in 71% of the test cases and achieves the same result in 17% of the test cases provided in the synthetic benchmarks. The experiments performed on DIMACS benchmarks denote that our algorithm finds the best number of colors in 70 out of 73 graphs, so the proposed work is very successful in coloring vertex-weighted graphs within a reasonable amount of time.

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“…For (H 1 , H 2 )-free graphs, the classification of Colouring is open for many pairs of graphs H 1 , H 2 . A summary of the known results can be found in [88], but several other results have since appeared [15,43,68,85,119,120,146]; see [68] for further details. In relation to boundedness of clique-width, the following is of importance.…”

Section: Graph Isomorphismmentioning

confidence: 99%

Clique-width for hereditary graph classes

Dabrowski¹,

Johnson²,

Paulusma³

2019

Surveys in Combinatorics 2019

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Polynomial Cases for the Vertex Coloring Problem

Cited by 19 publications

References 35 publications

Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Integrated Crossover Based Evolutionary Algorithm for Coloring Vertex-Weighted Graphs

Clique-width for hereditary graph classes

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