Colouring perfect graphs with a bounded number of colours

It follows from the ellipsoid method and results of Grotschel, Lovasz and Schrijver that one can find an optimal colouring of a perfect graph in polynomial time. But no ''combinatorial'' algorithm to do this is known.

Here we give a combinatorial algorithm to do this in an n-vertex perfect graph in time O(n^{k+1}^2) where k is the clique number; so polynomial-time for fixed k. The algorithm depends on another result, a polynomial-time algorithm to find a ''balanced skew partition'' in a perfect graph if there is one.

Joint work with Maria Chudnovsky, Aurelie Lagoutte, and Sophie Spirkl.