  Given a graph $H$, let $G^j_k(H)$ be the graph whose vertices are the proper
$k$-colorings of $H$, with edges joining two colorings if $H$ contains a
connected subgraph on at most $j$ vertices that includes all vertices where the
colorings differ. Properties of $G^1_k(H)$ have been investigated before,
including connectedness and Hamiltonicity. We introduce and study the
parameters $g_k(H)$ and $h_k(H)$, which denote the minimum $j$ such that
$G^j_k(H)$ is connected or Hamiltonian, respectively.
