Circular colorings of weighted graphs
Walter A. Deuber, Xuding Zhu · Journal of Graph Theory · 1996
Suppose that G is a finite 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 Δ of the vertices of G to arcs of C such that Δ(x)∩Δ(y) = 0 if (x, y) ∈ E(G) and Δ(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 fractional clique weight of (G, w) for arbitrary weight function w. We call such graphs star-superperfect. We prove that odd cycles and their complements are star-superperfect. We then prove a theorem about the circular chromatic number of lexicographic product of graphs which provides a tool of constructing new star-superperfect graphs from old ones. © 1996 John Wiley & Sons, Inc.