Chudnovsky, MariaLagoutte, AurélieSeymour, PaulSpirkl, Sophie2022-08-122022-08-122017-01https://doi.org/10.1016/j.jctb.2016.09.006http://hdl.handle.net/10012/18515The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2016.09.006. © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver [9] from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a “combinatorial” polynomial-time algorithm that yields an optimal colouring of a perfect graph. A skew partition in G is a partition (A, B) of V(G) such that G[A] is not connected and G[B] is not connected, where G denotes the complement graph; and it is balanced if an additional parity condition on certain paths in G and G is satisfied. In this paper we first give a polynomial-time algorithm that, with input a perfect graph, outputs a balanced skew partition if there is one. Then we use this to obtain a combinatorial algorithm that finds an optimal colouring of a perfect graph with clique number k, in time that is polynomial for fixed k.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalcolouring algorithmperfect graphbalanced skew partitionArticle