Complexity of unique list colorability

Abstract

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 [M. Mahdian, E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295–305] gave a polynomial-time characterization of uniquely 2-list colorable graphs. Answering an open question from [M. Ghebleh, E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. 59 (2001) 307–318; M. Mahdian, E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295–305], we show that uniquely 3-list colorable graphs are unlikely to have such a nice characterization, since recognizing these graphs is Σ 2 p -complete.