Colouring of graphs with Ramsey-type forbidden subgraphs

Abstract

A colouring of a graph $G=(V,E)$ is a mapping $c:V\to \{1,2,\dots \}$ such that $c(u)\ne c(v)$ if $uv\in E$; 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 $\mathcal{H}$, then it is called $\mathcal{H}$-free. The complexity of Colouring for $\mathcal{H}$-free graphs with $|\mathcal{H}|=1$ has been completely classified. When $|\mathcal{H}|=2$, 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 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.