The round-up property of the fractional chromatic number for proper circular arc graphs

Abstract

Let G = (V, E) be a graph and let k be a nonnegative integer. A vector c ∈ ℤ is called k-colorable iff there exists a coloring of G with k colors that assigns exactly c(v) colors to vertex v ∈ V. Denote by χ (G) and χf(G) the chromatic number and fractional chromatic number, respectively. We prove that χ(G) = ⌈χf(G)⌉ holds for every proper circular arc graph G. For this purpose, a more general round-up property is characterized by means of a polyhedral description of all k-colorable vectors. Both round-up properties are equivalent for proper circular arc graphs. The polyhedral description is established and, as a by-product, a known coloring algorithm is generalized to multicolorings. The round-up properties do not hold for the larger classes of circular arc graphs and circle graphs, unless P = NP. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 256–267, 2000