K4‐free graphs with no odd hole: Even pairs and the circular chromatic number

An odd hole in a graph is an induced cycle of odd length at least five. In this article we show that every imperfect K₄‐free graph with no odd hole either is one of two basic graphs, or has an even pair or a clique cutset. We use this result to show that every K₄‐free graph with no odd hole has circular chromatic number strictly smaller than 4. We also exhibit a sequence {H_n} of such graphs with lim_n→∞χ_c(H_n)=4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 303–322, 2010