Nice colourings and the 4-colour theorem
  Proving for triangulations an extended version of the 4-colour theorem by induction, we manage to exclude the case which led to the failure of Kempe's attempted proof. The new idea is to claim the existence of a "nice" 4-colouring, in which existing Kempe chains satisfy a special condition, and to show that the assumption of its non-existence by a counterexample always leads to a contradiction: hence there is such a colouring. The focus of this paper is not to find in the induction step a reducible colouring of the neighbourhood of a 5-valent vertex among all possible colourings -- which is known to be impossible - but to prove the mere existence of such a "nice" colouring among all possible colourings. 
Peter D\"orre