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Colouring perfect graphs with bounded clique number

Author(s): Chudnovsky, Maria; Lagoutte, Aurelie; Seymour, Paul D.; Spirkl, Sophie

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Abstract: A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grotschel, Lovasz, 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. (C) 2016 Elsevier Inc. All rights reserved.
Publication Date: Jan-2017
Electronic Publication Date: 28-Sep-2016
Citation: Chudnovsky, Maria, Lagoutte, Aurelie, Seymour, Paul, Spirkl, Sophie. (2017). Colouring perfect graphs with bounded clique number. JOURNAL OF COMBINATORIAL THEORY SERIES B, 122 (757 - 775. doi:10.1016/j.jctb.2016.09.006
DOI: doi:10.1016/j.jctb.2016.09.006
ISSN: 0095-8956
EISSN: 1096-0902
Pages: 757 - 775
Type of Material: Journal Article
Version: Author's manuscript

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