On Circular Chromatic Number And Circular Perfect Graph

The circular chromatic number of a graph G is a natural generalization of the chromatic number of a graph, but it contains more information about the structure of the graph G than x(G) does, since Vince'36^ carried out his first paper on this subject in 1988, many intersting results have been recieved[4, 5, 8, 11, 14-18, 20, 24, 27, 29, 31, 33-45]. In 1999, X.Zhu introduced the conception of circular perfect graph and gave a preliminary research[6, 7]. In this paper, we prove the following theorem with a completely new method: Suppose G is a triangle free graph, and for every vertex x of G, N[x] is a perfect graph, and Hx is a bipartite graph which contains no induced P4, then G is a circular perfect graph, where N[x] = {y:y ~ x,y V}U{x}, Hx is a subgraph of G induced by G - N[x].At the same time we have the following collary: Suppose G is a triangle free core graph, then G is Gkd ( k = 3d - 1 or k = 3d - 2 ) if and only if for every vertex x of G, N[x\ is perfect graph and Hx is a bipartite graph which contains no induced P4.