  For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which
is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In
this paper we provide a polynomial-time algorithm to test if a graph with no
induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the
list version of this problem, where each vertex is assigned a list of possible
colors, which is a subset of $\{1,2,3\}$.
