  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 uniquely
k-list 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.
