Abstract

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.