Some bounds for the b-chromatic number of a graph

Kouider, Mekkia; Mahéo, Maryvonne

doi:10.1016/s0012-365x(01)00469-1

Cited by 95 publications

(35 citation statements)

References 3 publications

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“…We have further shown that deciding b(G) = ∆ + 1 is NP-complete even for bipartite graphs, thus strengthening the results of [7,10]. Finally, we have determined the b-chromatic number of the random graph, almost surely.…”

Section: Resultssupporting

confidence: 81%

“…However, b-colorbility defines a problem more difficult than ordinary colorability. We prove that the problem to decide whether there is a b-coloring by t(G) colors is NP -complete even for connected bipartite graphs and t(G) = ∆(G) + 1, thus strengthening NP-completeness results of [7,10]. This result shows that already for bipartite graphs, b-colorability becomes a difficult problem.…”

Section: Proposition 1 ([67]) For Every Graph G χ(G) ≤ B(g) ≤ T(g)supporting

confidence: 67%

“…We strengthen some complexity results of [7] and present several new bounds on the b-chromatic number, including a tight (upto a small multiplicative constant) bound on the b-chromatic number of the random graph. To state the results we need to recall the upper bound presented already in [6].…”

Section: Introductionsupporting

confidence: 62%

“…Kouider and Mahéo [7] noticed that there are graphs which have a b-coloring with k colors, no b-coloring with k +1 but again a b-coloring with k +2 colors. In fact, the gap between the integers for which a b-coloring exists may be arbitrarily large.…”

Section: Lemma 1 Let G Be a Connected Bipartite Graph With Bipartitimentioning

confidence: 99%

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On the b-Chromatic Number of Graphs

Kratochvı́l

Tuza

Voigt

2002

Lecture Notes in Computer Science

106

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Abstract. The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi adjacent to at least one vertex of each Xj, j = i. We discuss on the tightness of some bounds on b(G) and on the complexity of determining b(G). We also determine the asymptotic behavior of b(Gn,p) for the random graph, within the accuracy of a multiplicative factor 2 + o(1) as n → ∞.

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Section: Resultssupporting

confidence: 81%

Section: Proposition 1 ([67]) For Every Graph G χ(G) ≤ B(g) ≤ T(g)supporting

confidence: 67%

Section: Introductionsupporting

confidence: 62%

Section: Lemma 1 Let G Be a Connected Bipartite Graph With Bipartitimentioning

confidence: 99%

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On the b-Chromatic Number of Graphs

Kratochvı́l

Tuza

Voigt

2002

Lecture Notes in Computer Science

106

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“…The b-chromatic number of the cartesian product of some graphs was studied in [16,18]. In particular, R. Javadi and B. Omoomi [16] showed that the b-chromatic number of K 3 K 3 is equal to 3.…”

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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A Distributed Algorithm for a b-Coloring of a Graph

Effantin

Kheddouci

2006

Parallel and Distributed Processing and Applications

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A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. Such a coloring gives a partitioning of the graph in clusters for which every cluster has a clusterhead (the dominating node) adjacent to each other cluster. Such a decomposition is very interesting for large distributed systems, networks,... In this paper we present a distributed algorithm to compute a b-coloring of a graph, and we propose an application for the routing in networks to illustrate our algorithm.

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Some bounds for the b-chromatic number of a graph

Cited by 95 publications

References 3 publications

On the b-Chromatic Number of Graphs

On the b-Chromatic Number of Graphs

On b-vertex and b-edge critical graphs

A Distributed Algorithm for a b-Coloring of a Graph

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