  For an integer $k \geq 1$ and a graph $G$, let $\mathcal{K}_k(G)$ be the
graph that has vertex set all proper $k$-colorings of $G$, and an edge between
two vertices $\alpha$ and~$\beta$ whenever the coloring~$\beta$ can be obtained
from $\alpha$ by a single Kempe change. A theorem of Meyniel from 1978 states
that $\mathcal{K}_5(G)$ is connected with diameter $O(5^{|V(G)|})$ for every
planar graph $G$. We significantly strengthen this result, by showing that
there is a positive constant $c$ such that $\mathcal{K}_5(G)$ has diameter
$O(|V(G)|^c)$ for every planar graph $G$.
