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For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is amapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever$uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a$k$-colouring of $G$ exists. The $k$-Precolouring Extension problem is todecide, for a given $G=(V,E)$, whether a colouring of a subset of $V$ can beextended to a $k$-colouring of $G$. A $k$-list assignment of a graph is anallocation of a list -a subset of $\{1,\ldots,k\}$- to each vertex, and theList $k$-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. Wecontinued the study of the computational complexity of these three decisionproblems when restricted to graphs that contain neither a cycle on $s$ verticesnor a path on $t$ vertices as induced subgraphs (for fixed positive integers$s$ and~$t$).

Narrowing the Complexity Gap for Colouring ($C_s$,$P_t$)-Free Graphs