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Complexity of unique list colorability. (English) Zbl 1146.05314
Summary: 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. M. Mahdian and E. S. Mahmoodian [A characterization of uniquely 2-list colorable graphs, Ars Combin. 51, 295–305 (1999; Zbl 0977.05046)] gave a polynomial-time characterization of uniquely 2-list colorable graphs. Answering an open question from M. Ghebleh and E. S. Mahmoodian [On uniquely list colorable graphs, Ars Comb. 59, 307–318 (2001; Zbl 1066.05063)] and M. Mahdian and E. S. Mahmoodian [loc. cit.], we show that uniquely 3-list colorable graphs are unlikely to have such a nice characterization, since recognizing these graphs is \(\Sigma_2^p\)-complete.

MSC:
05C85 Graph algorithms (graph-theoretic aspects)
05C15 Coloring of graphs and hypergraphs
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68R10 Graph theory (including graph drawing) in computer science
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