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Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chainis a subset of $V$ that induces a maximal connected subgraph of $G$ in whichevery vertex has one of two colours. To make a Kempe change is to obtain onecolouring 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 aseries 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 Kempeequivalent. We address the case $k=3$ by showing that all $3$-colourings of acubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ orthe triangular prism.

Kempe Equivalence of Colourings of Cubic Graphs