  The $k$-Colouring problem is to decide if the vertices of a graph can be
coloured with at most $k$ colours for a fixed integer $k$ such that no two
adjacent vertices are coloured alike. If each vertex u must be assigned a
colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain
the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain
$H$ as an induced subgraph. We continue an extensive study into the complexity
of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint
union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that
List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and
for $(P_3+P_4)$-free graphs. Combining our results with known results yields
complete complexity classifications of $3$-Colouring and List $3$-Colouring on
$H$-free graphs for all graphs $H$ up to seven vertices.
