Geodesic
Upper bounds on the b-chromatic number and results for restricted graph classes
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 709-735

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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.
Keywords: coloring, b-coloring, b-chromatic number, b-continuity 
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Alkhateeb, Mais; Kohl, Anja. Upper bounds on the b-chromatic number and results for restricted graph classes. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 709-735. http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a6/