List colourings of planar graphs

A graph G = G( V, E) is called L- list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L( v) associated with this vertex. We say G is k-choosable if all lists L( v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erdős, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable. We will prove the second conjecture.

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