Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k ＞ Δ, a Δ-colouring of G can be obtained by a sequence of O(n~2) recolourings using only the original k colours unless 1. G is a complete graph or a cycle with an odd number of vertices, or 2. k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours 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 exactly one vertex. It is known that 3. if k ≤ Δ(G), then R_k(G) might not be connected and it is possible that its connected components have superpolynomial diameter, 4. if k ≥ Δ(G) + 2, then R_k(G) is connected and has diameter O(n~2). We complete this structural classification by settling the missing case: 5. if k - Δ(G) + 1, then R_k(G) consists of isolated vertices and at most one further component which has diameter O(n~2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R_k(G) by settling the case k = Δ(G) + 1. The problem is 6. O(n~2) time solvable for k = 3, 7. PSPACE-complete for 4 ≤ k ≤ Δ(G), 8. O(n) time solvable for k = Δ(G) + 1, 9. O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes).
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