Claw-free circular-perfect graphs

The circular chromatic number of a graph is a well-studied refinement of the chromatic number. Circular-perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This paper studies claw-free circular-perfect graphs. First we prove that ifG is a connected claw-free circular-perfect graph with χ(G) > ω(G), then min{α(G), ω(G)} = 2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw-free circular-perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)} = 2. In contrast to this result, it is shown in [10] that minimal circularimperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this paper, we prove that claw-free minimal circular-imperfect graphs G have min{α(G), ω(G)} ≤ 3.