Circular perfect graphs

Abstract

AbstractFor 1 ≤ d ≤ k, let Kk/d be the graph with vertices 0, 1, …, k − 1, in which i ∼j if d ≤ |i − j| ≤ k − d. The circular chromatic number χc(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to Kk/d. The circular clique number ωc(G) of G is the maximum of those k/d for which Kk/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χc(H) = ωc(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, NG[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, NG[x] is a perfect graph and G − N[x] is a bipartite graph with no induced P5 (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj́os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph Kk/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005