  We consider the following type of question: Given a partial proper $d$-edge
coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors
for the non-colored edges of $Q_d$,can we extend the partial coloring to a
proper $d$-edge coloring using only colors from the lists? We prove that this
question has a positive answer in the case when both the partial coloring and
the color lists satisfy certain sparsity conditions.
