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On uniquely list colorable graphs. (English) Zbl 1066.05063
Summary: 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 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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