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) 6= 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 nonedge. 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 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.