Choosability of planar graphs (Q1916256)

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).