Abstarct: The purpose of this thesis is to consider one state of coloring the graphs named b-coloring. This coloring partitions the vertexes of graph into independent sets named color classes in such away which there is a vertex in each classes. That vertex possesses a neighbor in rest of classes. In following we have considered the biggest partition for particular graphs such as canned ,regular and the graphs which are derived from the product of tow Cartesian graphs.