A relation between choosability and uniquely list colorability

Let G be a graph with n vertices and m edges and assume that f : V ( G ) → N is a function with ∑ v ∈ 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 ∑ v ∈ V ( G ) f ( v ) > m + n , then there is no list assignment L such that | L v | = f ( v ) for any v ∈ 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 = u v , | L e | = max { d ( u ) , d ( v ) } , then G is not uniquely L-colorable and we conjecture that this result holds for any graph.