On The Complete B-coloring And The B-edge Coloring Of A Graph

The problem of coloring and coloring number are a important content and a very active field of the graph theory.It has very profound and plentiful theoretical result and extensive practical application.The theory and method of coloring in discrete mathematics occupies an important position.The b-chromatic number of a graph was introduced by Irving and Manlove in 1999. They have shown that determination of b(G) is NP-hard for general graphs, but polynomial for tree.The b-continuity of a graph was introduced by Faik in 2004.He prove that the chordal graph are b-continuous.The main works in this paper as follows:(1)The problem of b-coloring of the square grid graph was studied,the b-chromatic number of it was given,and it is proved that the square grid graph is b-continuous.(2)The concept of the complete b-chromatic number and the complete b-continuity of a graph is proposed,and the complete b-chromatic number of path,circle,square grid and fully n-ary tree graphs are studied,and it is proven that all these graphs are b-continuous in the paper.(3)The concept of the b-edge chromatic number and the b-edge continuity of a graph is proposed, and the b-edge chromatic number of path,circle,square grid and fully n-ary tree graphs are given, and it is proven that path,circle and fully n-ary tree graphs are b-edge continuous in the paper.(4)The concept of the complete b-edge chromatic number and the complete b-edge continuity of a graph is proposed,and the complete b-edge chromatic number of path,circle and fully n-ary tree graphs are given, and it is proven that all these graphs are complete b-edge continuous in the paper.