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Colouring of graphs with Ramsey-type forbidden subgraphs.

Dabrowski, K.K. and Golovach, P.A. and Paulusma, D. (2014) 'Colouring of graphs with Ramsey-type forbidden subgraphs.', Theoretical computer science., 522 . pp. 34-43.


A colouring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k -colouring. The Colouring problem is that of testing whether a given graph has a k -colouring for some given integer k . If a graph contains no induced subgraph isomorphic to any graph in some family H, then it is called H-free. The complexity of Colouring for H-free graphs with |H|=1 has been completely classified. When |H|=2, the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H1,H2}, where we allow H1 to have a single edge and H2 to have a single non-edge. Instead of showing only polynomial-time solvability, we prove that Colouring on such graphs is fixed-parameter tractable when parameterized by |H1|+|H2|. As a by-product, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.

Item Type:Article
Keywords:Colouring, Independent set, Clique, Forbidden induced subgraphs.
Full text:(AM) Accepted Manuscript
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Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in Theoretical computer science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Theoretical computer science, 522, 2014, 10.1016/j.tcs.2013.12.004
Date accepted:No date available
Date deposited:08 January 2015
Date of first online publication:February 2014
Date first made open access:No date available

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