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Let $G$ be a graph with $n$ vertices and $m$ edges and assume that
$f:V(G)\rightarrow N$ is a function with $\sum_{v\in
V(G)}f(v)=m+n$. We show that, if we can assign to any vertex $v$
of $G$, a list $L_{v}$ of size $f(v)$, such that $G$ has a unique
vertex coloring with these lists, then $G$ is $f$-choosable. This
implies that, if $\sum_{v\in V(G)}f(v)=m+n$, then there is no list
assignment $L$ such that $|L_{v}|=f(v)$ for any $v\in V(G)$, and
$G$ is uniquely $L$-colorable. Finally, we prove that if $G$ is a
connected non-regular multigraph with a list assignment $L$ of
edges, such that for each edge $e=uv,|L_{e}|=max\{d(u),d(v)\}$,
then $G$ is not uniquely $L$-colorable and we conjecture that this
result holds for any graph.
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