Let k and d be positive integers such that $k\ge d+2$. 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 colouring 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, there is no proof of a polynomial upper bound on the diameter, 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 $k\ge d+2$. Moreover, we prove that if $k\ge \frac{3}{2}(d+1)$ then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of $k\ge 2d+1$. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

令k和d为正整数，使得$ķ\ge d+2$. 考虑d简并图G的两种k着色。我们是否可以通过在每一步重新着色一个顶点，同时在任何一步保持适当的着色来将一个顶点转换为另一个？塞雷塞达等人。肯定地回答了这个问题，并展示了一个指数长度的重新着色序列。然而，Cereceda 推测应该存在一个二次长度。

G的k重构图是其顶点是G的正确k着色的图，如果它们在一个顶点上不同，则在两种着色之间有一条边。Cereceda 的猜想可以重新表述如下：$(d+2)$-n顶点上的任何d退化图的重新配置图是$○(n^2)$. 到目前为止，还没有证明直径的多项式上限，即使对于$d=2$.

在本文中，我们证明了d-退化图的k-重构图的直径为$○(n^{d+1})$为了$ķ\ge d+2$. 此外，我们证明如果$ķ\ge \frac{3}{2}(d+1)$那么k -reconfiguration 图的直径是二次的，改进了之前的界限$ķ\ge 2d+1$. 我们还表明，平面二部图的 5 重配置图具有二次直径，证实了 Cereceda 对此类图的猜想。