The Investigation About The Property Of Graph G_d～k

In this paper,we will study two kinds of graphs G and C which almost complement with each other. The main part of this paper study the properties of graphs G,and the study of graphs C appears in the discussion.Since 1988 Vince proposed the conception of star-chromatic number, a large amount of study about star-chromatic number, circular chromatic number and the way to do study have been achieved.Then graph G were produced.lt has vertex set {0,1,2,..., (k-1)} and i ~ j if and only if . If we study star-chromatic number in view of homomorphism, the graphs G play the same role that complete graphs Kn do in the study of chromatic numbers, that is: G is n-colorable if and only if G horno-inorphic to Kn. For every rational number 2, there exist a special G^ graphs.such that: G is |-colorable if and only if G homomorphic to G1.In the course of study, graphs G not only have so important role,but also have many good properties. In this paper, we will study some properties of graph G.In graph theory, coloring problem and chromatic number problem are very active study topic. Some core problems caused many people's interests,but they have not been resolved. Such as the Four color-theory of planar.In this paper, we will obtain a conclusion: Graphs G(k > 2d,gcd(k,d) = 1) other than odd circle are nonplanar and then based on it, we easily obtain two corollaries about circular clique number(clique number) and circular chromatic number (chromatic number) of planar: For any planar, it's circular clique number is 2 4- or integers not more than 4; If a planar is circular perfect or (perfect), then it satisfy Four color-theory.