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 $c: V\rightarrow{1,2,\ldots,k}$ 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 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 ${1,\ldots,k}$- 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 contain neither a cycle on $s$ vertices nor a path on $t$ vertices as induced subgraphs (for fixed positive integers $s$ and~$t$).