### Abstract

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 language | English (US) |
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Pages (from-to) | 757-775 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 122 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Balanced skew partition
- Colouring algorithm
- Perfect graph

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*122*, 757-775. https://doi.org/10.1016/j.jctb.2016.09.006