On the b-dominating coloring of graphs

Hoíng, Chınh T.; Kouider, Mekkia

doi:10.1016/j.dam.2005.04.001

Cited by 47 publications

(30 citation statements)

References 6 publications

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“…Let P, F 1 , F 2 and K 3,3 be the graphs depicted in the Figure 1. The next result was established in [12].…”

Section: Some Known Resultsmentioning

confidence: 99%

On b-vertex and b-edge critical graphs

Eschouf¹,

Blidia²

2015

OpMath

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Abstract. A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b + -vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In this note, we explain some properties in b + -vertex (edge) critical graphs, and we conclude with two open problems.

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“…Let P, F 1 , F 2 and K 3,3 be the graphs depicted in the Figure 1. The next result was established in [12].…”

Section: Some Known Resultsmentioning

confidence: 99%

On b-vertex and b-edge critical graphs

Eschouf¹,

Blidia²

2015

OpMath

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“…, j − 1}. Every member of X has a neighbor of color g, so, by (8), there is a vertex v g of color g that is adjacent to all of X. Note that v g is in Z ∪ T .…”

Section: Now We Claim Thatmentioning

confidence: 99%

“…. , j − 1, vertex x i has a neighbor of color j, so, by (8), there exists a vertex u j of color j that is adjacent to all of X; moreover u j is in Z since color j does not appear in T . If u j is a b-vertex, then the desired conclusion holds with clique X ∪ {u j }.…”

Section: Now We Claim Thatmentioning

confidence: 99%

“…However, graphs such that χ(G) = b(G) do not have a specific structure; to see this, we can take any arbitrary graph G and add a component that consists of a clique of size b(G); we obtain a graph G ′ that satisfies χ(G ′ ) = b(G ′ ) = b(G). This led Hoàng and Kouider [8] to introduce the class of b-perfect graphs: a graph G is called b-perfect if every induced subgraph H of G satisfies χ(H) = b(H). Hoàng and Kouider [8] proved the b-perfectness of some classes of graphs, and asked for a good characterization of the whole class of b-perfect graphs.…”

Section: Introductionmentioning

confidence: 99%

“…This led Hoàng and Kouider [8] to introduce the class of b-perfect graphs: a graph G is called b-perfect if every induced subgraph H of G satisfies χ(H) = b(H). Hoàng and Kouider [8] proved the b-perfectness of some classes of graphs, and asked for a good characterization of the whole class of b-perfect graphs. Hoàng, Linhares Sales and Maffray [9] proposed the conjecture below.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Characterization of b‐Perfect Graphs

Hoàng

Maffray

Mechebbek

2011

Journal of Graph Theory

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A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b-perfect if and only if it does not contain as an induced subgraph a member of a certain list of twenty-two graphs. This entails the existence of a polynomial-time recognition algorithm and of a polynomial-time algorithm for coloring exactly the vertices of every b-perfect graph.

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b-Coloring is NP-Hard on Co-Bipartite Graphs and Polytime Solvable on Tree-Cographs

Bonomo

Schaudt

Stein

et al. 2014

Lecture Notes in Computer Science

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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 by χ b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ(G), . . . , χ b (G), and 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 .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 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.

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On the b-dominating coloring of graphs

Cited by 47 publications

References 6 publications

On b-vertex and b-edge critical graphs

On b-vertex and b-edge critical graphs

A Characterization of b‐Perfect Graphs

b-Coloring is NP-Hard on Co-Bipartite Graphs and Polytime Solvable on Tree-Cographs

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