Circular perfect graphs

• Authors: Xuding Zhu
• Venue: Journal of Graph Theory
• Year: 2005

For 1 ≤ d ≤k, letKk-d be the graph with vertices 0,1,…, k - 1, in which i j ifd ≤ |i - j| ≤k - d. Thecircular chromatic number χc(G) of agraph G is the minimum of those k-d for whichG admits a homomorphism to Kk-d.The circular clique number ωc(G) ofG is the maximum of those k-d for whichKk-d admits a homomorphism to G. Agraph G is circular perfect if for every induced subgraphH of G, we have χc(H)=ωc(H). In this paper, we prove thatif G is circular perfect then for every vertex x ofG, NG[x] is a perfect graph.Conversely, we prove that if for every vertex x of G,NG[x] is a perfect graph andG - N[x] is a bipartite graph with no inducedP5 (the path with five vertices), then Gis a circular perfect graph. In a companion paper, we apply themain result of this paper to prove an analog of Hajos theorem forcircular chromatic number for k-d ≥ 3. Namely, we shalldesign a few graph operations and prove that for any k-d≥ 3, starting from the graph Kk-d, onecan construct all graphs of circular chromatic number at leastk-d by repeatedly applying these graph operations. ©2005 Wiley Periodicals, Inc. J Graph Theory 48: 186209, 2005