Complexity of unique list colorability

Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that |L(v)| = k for every vertex v and the graph has exactly one L-coloring with these lists. Mahdian and Mahmoodian [MM99] gave a polynomial-time characterization of uniquely 2-list colorable graphs. Answering an open question from [GM01,MM99], we show that uniquely 3-list colorable graphs are unlikely to have such a nice characterization, since recognizing these graphs is Σp2-complete.