Given a graph G, b-coloring is a proper fc-coloring of G in which every color class has at least one vertex that has a neighbour in each of the other color classes. Such a vertex is called a b-vertex. A set S0 ⊆ V is called a b-system if all the vertices in S0 are b-vertices that belong to different color classes. The b-chromatic number is the largest integer κ such that G admits a b- coloring with κ colors. AJ greater b-chromatic graph is a graph G such that the achromatic number of G is greater than the b-chromatic number of every graph obtained from G by identifying two adjacent vertices, denoted by bh,-chromatic. In this paper, for any given graph G we constructed the graphs G∗, G+ and G∗+. It is proved thatjG∗, G+ axe bh-chromatic graph but G∗+ is noi bh-chromatic graph. Further, we proved that the b-chromatic number of G∗+ is neither equal nor less than the b-chromatic number of every graph obtained from G∗+ by identifying any two adjacent vertices. © 2017 Allahabad Mathematical Society.