The dominating coloring of a graph G, i.e., proper coloring in wich for each color i there exist a vertex x i of color i such that for every color j ?i, there exists a vertex x j of color j adjacent to x i . The largest number k such that G has dominating coloring with k colors is saied b-chromatic number and denoted by j(G). These concepts were Introduced by Irving and Manlove (1999). The b-chromatic number of some graphs have been determined and some different bounds have been given for j(G) up to now. The second and third chapters of this thesis contains of these results. – A graph G is b-perfect if each induced subgrah H of G has j(H) = (H), where (H) is the chromatic number of H. Hoang and Kouider (2005) characterized all b-perfect bipartite graphs and all P 4 –sparse b-perfect graphs. They also proved that every 2K 2 –free and p 5 –free graph is b-perfect. We peresnt these results in chapter fourth.