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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 ofsize $1$ and $s$ respectively. The one-subdivision of $K_{1,s}$ is obtained byreplacing 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-timealgorithm for the list-three-coloring problem restricted to the class of$P_t$-free graph with no induced 1-subdivision of $K_{1,s}$.

List-three-coloring $ P_t $-free graphs with no induced 1-subdivision of $ K_{1,s} $