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A graph is said 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.
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