  A graph is called to be uniquely list colorable, if it admits a list
assignment which induces a unique list coloring. We study uniquely list
colorable graphs with a restriction on the number of colors used. In this way
we generalize a theorem which characterizes uniquely 2-list colorable graphs.
We introduce the uniquely list chromatic number of a graph and make a
conjecture about it which is a generalization of the well known Brooks'
theorem.
