Circular colorings of weighted graphs

Suppose that G is a nite simple graph and w is a weight function which assigns to each vertex of G a nonnegative real number. Let C be a circle of length t. A t-circular coloring of (G; w) is a mapping of the vertices of G to arcs of C such that (x)\(y) = if xy 2 E (G) and (x) has length w(x). The circular-chromatic number of (G; w) is the least t for which there is a t-circular coloring of (G; w). This paper discusses basic properties of circular chromatic number of a weighted graph and relations between this parameter and other graph parameters. We are particularly interested in graphs G for which the circular-chromatic number of (G; w) is equal to the fractonal clique weight of (G; w) for arbirtary weight function w. We call such graphs star-superperfect. We prove that odd cycles and the complement of odd cycles are star-superperfect. We then consider circular chromatic number of lexicographic product of graphs, which provides a tool of constructing new star-superperfect graphs from old ones.