Colouring perfect graphs with bounded clique number

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.