A Reconfigurations Analogue of Brooks' Theorem and its Consequences
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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 an analogue of Brooks' Theorem by proving that from any 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 G is a complete graph or a cycle with an odd number of vertices, or 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. We prove that for Δ≥ 3, R_Δ+1(G) consists of isolated vertices and at most one further component which has diameter O(n^2). This result enables us to complete both a structural classification and an algorithmic classification for reconfigurations of colourings of graphs of bounded maximum degree.
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