Towards Cereceda's conjecture for planar graphs
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The reconfiguration graph R_k(G) of the k-colourings of a graph G has as vertex set the set of all possible k-colourings of G and two colourings are adjacent if they differ on the colour of exactly one vertex. Cereceda conjectured ten years ago that, for every k-degenerate graph G on n vertices, R_k+2(G) has diameter O(n^2). The conjecture is wide open, with a best known bound of O(k^n), even for planar graphs. We improve this bound for planar graphs to 2^O(√(n)). Our proof can be transformed into an algorithm that runs in 2^O(√(n)) time.
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