Narrowing the Complexity Gap for Colouring (C s , P t )-Free Graphs

For a positive integer k and graph G = (V,E), a k-colouring of G is a mapping c : V → {1, 2, . . . , k} such that c(u) 6= c(v) whenever uv ∈ E. The k-Colouring problem is to decide, for a given G, whether a k-colouring of G exists. The k-Precolouring Extension problem is to decide, for a given G = (V,E), whether a colouring of a subset of V can be extended to a k-colouring of G. 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 is to decide, for a given G, whether G has a k-colouring in which each vertex is coloured with a colour from its list. We continued the study of the computational complexity of 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).