Bounds for the $b$-chromatic number of some families of graphs
  In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight. 
Mekkia Kouider and Manouchehr Zaker