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Let $G$ be a simple undirected graph on $n$ vertices with maximumdegree~$\Delta$. Brooks' Theorem states that $G$ has a $\Delta$-colouringunless~$G$ is a complete graph, or a cycle with an odd number of vertices. Torecolour $G$ is to obtain a new proper colouring by changing the colour of onevertex. We show an analogue of Brooks' Theorem by proving that from any$k$-colouring, $k>\Delta$, a $\Delta$-colouring of $G$ can be obtained by asequence of $O(n^2)$ recolourings using only the original $k$ colours unless$G$ is a complete graph or a cycle with an odd number of vertices, or$k=\Delta+1$, $G$ is $\Delta$-regular and, for each vertex $v$ in $G$, no twoneighbours of $v$ are coloured alike.  We use this result to study the reconfiguration graph $R_k(G)$ of the$k$-colourings of $G$. The vertex set of $R_k(G)$ is the set of all possible$k$-colourings of $G$ and two colourings are adjacent if they differ on exactlyone vertex. We prove that for $\Delta\geq 3$, $R_{\Delta+1}(G)$ consists ofisolated vertices and at most one further component which has diameter$O(n^2)$. This result enables us to complete both a structural classificationand an algorithmic classification for reconfigurations of colourings of graphsof bounded maximum degree.

A Reconfigurations Analogue of Brooks' Theorem and its Consequences