On Uniquely List Colorable Graphs

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.