Geodesic
Bounds for the b-Chromatic Number of Subgraphs and Edge-Deleted Subgraphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 959-976

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A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain bounds for the b- chromatic number of induced subgraphs in terms of the b-chromatic number of the original graph. This turns out to be a generalization of the result due to R. Balakrishnan et al. [Bounds for the b-chromatic number of G−v, Discrete Appl. Math. 161 (2013) 1173-1179]. Also we show that for any connected graph G and any e ∈ E(G), b(G - e) ≤ b(G) + n/2 - 2. Further, we determine all graphs which attain the upper bound. Finally, we conclude by finding bound for the b-chromatic number of any subgraph.
Keywords: b-coloring, b-chromatic number 
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Francis, P.; Raj, S. Francis. Bounds for the b-Chromatic Number of Subgraphs and Edge-Deleted Subgraphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 959-976. http://geodesic.mathdoc.fr/item/DMGT_2016_36_4_a13/