Bounds for the b-chromatic number of some families of graphs

Kouider, Mekkia; Zaker, Manouchehr

doi:10.1016/j.disc.2006.01.012

Cited by 55 publications

(23 citation statements)

References 5 publications

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“…Suppose that x sees u 4 . If x misses v 4 , then {b, x, a 3 , u 4 , v 4 , a 5 } induces an F 17 , while if x sees v 4 , then the same set induces an F 10 . So x misses u 4 .…”

Section: Lemma 24 Let G Be a Minimal B-imperfect F -Free Graph Thementioning

confidence: 99%

“…We claim that Y is either a stable set or a clique. (10) For in the opposite case, Y contains three vertices y, y , y that induce a subgraph with either one edge or two edges. If it induces two edges, then {z, y, y , y } induces a diamond.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The NP-completeness results have incited researchers to establish bounds on the b-chromatic number in general or to find its exact values for subclasses of graphs (see [3,9,10,2,4,8]). Clearly every χ (G)-coloring of a graph G is a b-coloring, and so every graph G satisfies χ (G) ≤ b(G).…”

Section: Introductionmentioning

confidence: 99%

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On minimally b-imperfect graphs

Hoíng¹,

Sales²,

Maffray³

2009

Discrete Applied Mathematics

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a b s t r a c tA b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. 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 H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.

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“…Suppose that x sees u 4 . If x misses v 4 , then {b, x, a 3 , u 4 , v 4 , a 5 } induces an F 17 , while if x sees v 4 , then the same set induces an F 10 . So x misses u 4 .…”

Section: Lemma 24 Let G Be a Minimal B-imperfect F -Free Graph Thementioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

See 1 more Smart Citation

On minimally b-imperfect graphs

Hoíng¹,

Sales²,

Maffray³

2009

Discrete Applied Mathematics

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“…In [24], Kratochvíl et al show that determining χ b (G) is NP-hard even if G is a connected bipartite graph. More results on algorithmic aspects and bounds for some graph classes can be found in [3,7,8,16,23]. …”

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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“…The b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with exactly k colors. The concept of b-coloring was introduced in [6] and has been studied among others in [2,4,7,8,9]. Let ω(G) be the maximum size of a clique in a graph G, and let χ(G) be the chromatic number of G. It is easy to see that every coloring of G with χ(G) colors is a b-coloring, and so every graph satisfies χ(G) ≤ b(G).…”

Section: Introductionmentioning

confidence: 99%