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Some concepts in list coloring. (English) Zbl 1043.05046
In this paper the uniquely list colorable graphs and the list critical graphs are discussed. It is proved that every triangle-free uniquely \((k+1)\)-colorable graph is uniquely \(k\)-list colorable (Theorem 1), and every planar graph has \(m\)-number at most 4 (Theorem 5). Furthermore, all 3-list critical graphs are characterized (Theorem 7). It is conjectured that every \(\chi'_l\)-critical graph is \(\chi'\)-critical and the equivalence of this conjecture to the well-known list coloring conjecture is proved, which is very interesting and valuable.
By the way, the authors say that it seems that if \(f(v) = k\) for each vertex \(v\) of graph \(G\), the equality in Theorem 2 does not hold. But they do not provide the proof strictly.

05C15 Coloring of graphs and hypergraphs
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