List colourings of planar graphs
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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 Erdos, 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.
[1] L. Lovász. Combinatorial problems and exercises , 1979 .
[2] Noga Alon,et al. Colorings and orientations of graphs , 1992, Comb..