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 u is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(u) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erd&, 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] Barry A. Tesman. List T-Colorings of Graphs , 1993, Discret. Appl. Math..