 Let $(G,{\kappa_0})$ and $(H,{\kappa_0})$ be graph colorings in $k$ colors such that $(G,\kappa_0) \approx_{\pi,\sigma}   (H,\kappa_0)$. Then, $$(G,\kappa_0) \in {{\cal R}}(r_0,r_0,\ldots,r_k;n) \iff (H,\kappa_0) \in   {{\cal R}}(\sigma(r_0),\sigma(r_0),\ldots,\sigma(r_k);n)$$  