  Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain
is a subset of $V$ that induces a maximal connected subgraph of $G$ in which
every vertex has one of two colours. To make a Kempe change is to obtain one
colouring from another by exchanging the colours of vertices in a Kempe chain.
Two colourings are Kempe equivalent if each can be obtained from the other by a
series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$,
all $k$-colourings of $k$-regular graphs that are not complete are Kempe
equivalent. We address the case $k=3$ by showing that all $3$-colourings of a
cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or
the triangular prism.
