An r-circular coloring of a graph G is a map f from V(G) to the set of open unit intervals of an Euclidean circle of length r, such that f(u) ∩ f(v) = Ф whenever uv ∈ E(G). Circular perfect graphs are defined analogously to perfect graphs by means of two parameters, the circular chromatic number and the circular clique number. In this paper, we study the properties of circular perfect graphs. We give (1) a necessary condition for a graph to be circular perfect, (2) some circular critical imperfect graphs, and (3) a characterization of graphs with the property that each of their induced subgraphs has circular clique number the same as its clique number, and then the two conjectures that are equivalent to the perfect graph conjecture.
The circular clique number of a graph G is the maximum fractional k/d suchthat G_d^k admits a homomorphism to G. In this paper, we give some sufficient conditions for graphswhose circular clique number equal the clique number, we also characterize the K_(1,3)-free graphsand planar graphs with the desired property.
