Narrowing the Complexity Gap for Colouring (Cs, Pt)-Free Graphs

Let k be a positive integer. The k-Colouring problem is to decide whether a graph has a k-colouring. The k-Precolouring Extension problem is to decide whether a colouring of a subset of a graph's vertex set can be extended to a k-colouring of the whole graph. A k-list assignment of a graph is an allocation of a list - a subset of {1, ... , k} - to each vertex, and the List k-Colouring problem asks whether the graph has a k-colouring in which each vertex is coloured with a colour from its list. We prove a number of new complexity results for these three decision problems when restricted to graphs that do not contain a cycle on s vertices or a path on t vertices as induced subgraphs (for fixed positive integers s and t).