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Choosability of planar graphs. (English) Zbl 0852.05048
We say the graph $$G= (V, E)$$ is $$k$$-choosable if there is at least one $$L$$-list colouring for every possible list assignment $$L$$ with $$|L(v)|= k$$ for all $$v\in V$$. $$G$$ is called free $$k$$-choosable if such an $$L$$-list colouring exists for every list assignment $$L$$, every vertex $$v$$ and every colour $$f\in L(v)$$. It is shown that the following conjectures are equivalent: every planar graph is 5-choosable (free 5-choosable).

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
$$k$$-choosable; colouring; list assignment
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##### References:
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