  Let $k$ and $d$ be 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 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 $k \ge d+2$. Moreover, we prove that
if $k \ge \frac 32 (d+1)$ 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.
