  Let $k$ be an integer. Two vertex $k$-colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other
through a sequence of adjacent proper $k$-colorings. Any graph is
$(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two $(tw+2)$-colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
  Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum
degree plus two. We improve Jerrum's bound using the grundy number, which is
the worst number of colors in a greedy coloring.
