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.