A Reconfigurations Analogue of Brooks' Theorem and its Consequences

Let G be a simple undirected graph on n vertices with maximum degree ∆. Brooks’ Theorem states that G has a ∆-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 an analogue of Brooks’ Theorem by proving that from any k-colouring, k > ∆, a ∆-colouring of G can be obtained by a sequence of O(n) recolourings using only the original k colours unless – G is a complete graph or a cycle with an odd number of vertices, or – k = ∆ + 1, G is ∆-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 Rk(G) of the kcolourings of G. The vertex set of Rk(G) is the set of all possible kcolourings of G and two colourings are adjacent if they differ on exactly one vertex. We prove that for ∆ ≥ 3, R∆+1(G) consists of isolated vertices and at most one further component which has diameter O(n). This result enables us to complete both a structural classification and an algorithmic classification for reconfigurations of colourings of graphs of bounded maximum degree.