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On the complete b-chromatic number of a graph

A concept called the complete b-coloring(b-chromatic number) for vertices of graphs by strengthening the condition of the b-coloring(b-chromatic number) was proposed.In the problem of b-coloring,each color class has at least a b-coloring vertex.In the complete b-coloring,colorins not only must meet the above conditions,but also refer to each vertex v.The degree of vertex v is equal to or great than the complete b-chromatic number of graph G minus 1,and v is a b-coloring vertex.The problem of complete b-coloring of some graph is studied,such as paths,the circles,square grids and the full n-ary tree.The coloring schemes for paths,circles,square grids and the full n-ary tree are proposed,and the complete b-chromatic number of these graphs is obtained