Colouring perfect graphs with bounded clique number

Maria Chudnovsky, Aurélie Lagoutte, Paul Seymour, Sophie Spirkl

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)757-775
Number of pages19
JournalJournal of Combinatorial Theory. Series B
StatePublished - Jan 1 2017

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


  • Balanced skew partition
  • Colouring algorithm
  • Perfect graph


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