A polynomial version of Cereceda's conjecture
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Let k and d be such that k > d+2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d+2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n^2). So far, the existence of a polynomial diameter is open even for d=2. In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(n^d+1) for k > d+2. Moreover, we prove that if k >3/2 (d+1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k > 2d+1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.
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