On the b-Coloring of Cographs and P 4-Sparse Graphs

Bonomo, Flavia; Durán, Guillermo; Maffray, Frédéric; Marenco, Javier; Valencia-Pabon, Mario

doi:10.1007/s00373-008-0829-1

Cited by 46 publications

(48 citation statements)

References 13 publications

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“…In [12] (see also [6]) it is proved that chordal graphs and some planar graphs are b-continuous. A graph G is defined to be b-monotonic [3] if χ b (H 1 ) ≥ χ b (H 2 ) for every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1 . They proved that P 4 -sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic.…”

Section: Proposition 1 For Every Graph G χ(G) ≤ χ B (G) ≤ M(g)mentioning

confidence: 99%

“…The notion of dominance sequence has been introduced in [3] in order to compute the b-chromatic number of P 4 -sparse graphs and, in particular, cographs. Formally, given a graph G, the dominance sequence dom G ∈ Z N ≥ χ(G) , is defined such that dom G [t] is the maximum number of distinct color classes admitting dominant vertices in any coloring of G with t colors, for every t ≥ χ(G).…”

Section: Tree-cographsmentioning

confidence: 99%

“…Formally, given a graph G, the dominance sequence dom G ∈ Z N ≥ χ(G) , is defined such that dom G [t] is the maximum number of distinct color classes admitting dominant vertices in any coloring of G with t colors, for every t ≥ χ(G). Note that it suffices to consider this sequence until t = |V (G)|, since dom G The following results given in [3] are very important in order to compute the bchromatic number of graphs that can be decomposed recursively in modules via disjoint union or join operations.…”

Section: Tree-cographsmentioning

confidence: 99%

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$$b$$ b -Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

et al. 2014

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A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted byfor every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1 . We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with Algorithmica no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

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Section: Proposition 1 For Every Graph G χ(G) ≤ χ B (G) ≤ M(g)mentioning

confidence: 99%

Section: Tree-cographsmentioning

confidence: 99%

Section: Tree-cographsmentioning

confidence: 99%

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$$b$$ b -Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

et al. 2014

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“…It is clear that for a graph G to have a b-colouring of k colours, G must contain at least k vertices, each of degree at least k -1. Bonomo et al (2009) proved that P 4 -sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, they described a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.…”

Section: Introductionmentioning

confidence: 99%

The b-chromatic number of Mycielskian of some graphs

Lisna¹,

Sunitha²

2016

IJCONVC

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A b-colouring of a graph G is a proper colouring of the vertices of G such that there exists a vertex in each colour class joined to at least one vertex in each other colour classes. The b-chromatic number of a graph G, denoted by φ(G) is the largest integer k such that G has a b-colouring with k colours. The Mycielskian or Mycielski graph µ(H) of a graph H with vertex set {v 1 , v 2 , …, v n } is a graph G obtained from H by adding n + 1 new vertices {u, u 1 , u 2 , …, u n }, joining u to each vertex u i (1 ≤ i ≤ n) and joining u i to each neighbour of v i in H. In this paper, we obtain the b-chromatic number of Mycielskian of paths, complete graphs, complete bipartite graphs and wheels.

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“…A graph G is defined to be b-monotonic if χ b (H 1 ) ≥ χ b (H 2 ) for every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1 [3].…”

Section: Introductionmentioning

confidence: 99%

On the b-coloring of P4-tidy graphs

Velasquez

Bonomo

Koch

2011

Discrete Applied Mathematics

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a b s t r a c tA b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromaticsubgraph H 1 of G, and every induced subgraph H 2 of H 1 . In this work, we prove that P 4 -tidy graphs (a generalization of many classes of graphs with few induced P 4 s) are b-continuous and b-monotonic. Furthermore, we describe a polynomial time algorithm to compute the b-chromatic number for this class of graphs.

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On the b-Coloring of Cographs and P 4-Sparse Graphs

Cited by 46 publications

References 13 publications

$$b$$ b -Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

$$b$$ b -Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

The b-chromatic number of Mycielskian of some graphs

On the b-coloring of P4-tidy graphs

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