Let G be a graph with a proper vertex colouring. Let a and b be two of the colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. Another proper colouring of G can be obtained by swapping the colours on the vertices of a Kempe chain. This colouring is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.

Kempe chains were introduced by Alfred Kempe in an attempt to prove the Four Colour Theorem. Though this attempt failed, the method of Kempe chains was used in the proof of the Five Colour Theorem and elsewhere.

We address a 2007 conjecture of Bojan Mohar that asserts that, for k at least 3, all k-colourings of a k-regular graph that is not complete are Kempe equivalent. We show that the conjecture holds with one exception, the triangular prism.

This is joint work with Marthe Bonamy, Nicolas Bousquet, Carl Feghali and Daniel Paulusma.​