Abstract
Proving for triangulations an extended (sharper) 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 (a counterexample) always leads to a contradiction. The trick is to employ recolouring in order to assure the non-existence of potential counter-examples.