List colourings of planar graphs

AbstractA 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