Upper bounds on the b-chromatic number and results for restricted graph classes

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 ${\displaystyle {\chi }_b\left(G\right)}$ 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 ${\displaystyle \chi \left(G\right)\le k\le {\chi }_b\left(G\right)}$. In this paper, we establish four general upper bounds on ${\displaystyle {\chi }_b\left(G\right)}$. 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 ${\displaystyle {\chi }_b\left(G\right)}$ 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.