Colouring of graphs with Ramsey-type forbidden subgraphs

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 \({\cal H}\), then it is called \({\cal H}\)-free. The complexity of Colouring for \({\cal H}\)-free graphs with \(|{\cal H}|=1\) has been completely classified. When \(|{\cal H}|=2\), the classification is still wide open, although many partial results are known. We continue this line of research and forbid induced subgraphs {H 1,H 2}, where we allow H 1 to have a single edge and H 2 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 |H 1| + |H 2|. As a byproduct, we obtain the same result both for the problem of determining a maximum independent set and for the problem of determining a maximum clique.