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Let $G$ be a graph with $n$ vertices and suppose that for each vertex $v$ in $G$, there exists a list of $k$ colors, $L(v)$, such that there is a unique proper coloring for $G$ from this collection of lists, then
$G$ is called a {\it uniquely} $k$-list {\it colorable graph}. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2-list
colorable graphs. Here we state some results which will pave the way in characterization of uniquely $k$-list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares.
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