Every planar graph is 4-colourable - two proofs without computer
Peter Doerre · 2004
Colouring of planar graphs can be treated as a special list-colouring problem with selected lists for near-triangulations. The new idea is to use sublists of a common list of four colours, to enforce a common colour in all lists, and to admit on the bounding cycle at most one vertex with a list of at least two colours. By these conditions colours and lists are excluded, which have to be considered in list-colouring (leading to the well-known result that planar graphs are 5-choosable). The essential application is a proof of the 4-colour theorem.