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) + [...] -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.
A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number $χ_b(G)$ is the maximum integer k for which G has a b-coloring by k colors. Moreover, the graph G is called b-continuous if G admits a b-coloring by k colors for all k satisfying $χ(G) ≤ k ≤ χ_b(G)$. In this paper, we establish four general upper bounds on $χ_b(G)$. We present results on the b-chromatic number and the b-continuity problem for special graphs, in particular for disconnected graphs and graphs with independence number 2. Moreover we determine $χ_b(G)$ for graphs G with minimum degree δ(G) ≥ |V(G)|-3, graphs G with clique number ω(G) ≥ |V(G)|-3, and graphs G with independence number α(G) ≥ |V(G)|-2. We also prove that these graphs are b-continuous.
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