  A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex
$v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from
$L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color
sets. In this paper, we give necessary and sufficient conditions for a weighted
path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we
solve the problem of the free-choosability of a cycle.
