Let $G = (V, E)$ be a $d$-regular graph, drawn randomly using the configuration model. The number of vertices is $|V| = n$.
Under this model, we know that the graph looks tree-like locally. In other words, we know that if $X_r$ were the number of vertices in $G$ whose $r$-neighborhood is not a tree, then
$\mathbb{E}[X_r] = o(n)$ for $r \leq \frac{\log_{d-1}n}{2}$.
Only a constant fraction of vertices contain cycles in their $r$-neighborhoods.
See here for instance. (Slide 19 of lecture notes from a course at MIT)
I would like to know whether I can partition this graph in such a way that each partition is a tree, the partitions do not overlap and together constitute the entire graph? If not, can I do something close to this or perhaps approximate the original graph this way?
Thanks.