Measurable domatic partitions

Let $\Gamma$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma$, a domatic $\aleph_0$-partition (for its Schreier graph on $\Gamma$) is a partial function $f:\Gamma\rightharpoonup\mathbb{N}$ such that for every $x\in \Gamma$, one has $f[S\cdot x]=\mathbb{N}$. We show that a continuous domatic $\aleph_0$-partition exists, if and only if a Baire measurable domatic $\aleph_0$-partition exists, if and only if the topological closure of $S$ is uncountable. A Haar measurable domatic $\aleph_0$-partition exists for all choices of $S$. We also investigate domatic partitions in the general descriptive graph combinatorial setting.