  Let $s$ and $t$ be positive integers. We use $P_t$ to denote the path with
$t$ vertices and $K_{1,s}$ to denote the complete bipartite graph with parts of
size $1$ and $s$ respectively. The one-subdivision of $K_{1,s}$ is obtained by
replacing every edge $\{u,v\}$ of $K_{1,s}$ by two edges $\{u,w\}$ and
$\{v,w\}$ with a new vertex $w$. In this paper, we give a polynomial-time
algorithm for the list-three-coloring problem restricted to the class of
$P_t$-free graph with no induced 1-subdivision of $K_{1,s}$.
