  A vertex-coloring of a hypergraph is conflict-free, if each edge contains a
vertex whose color is not repeated on any other vertex of that edge. Let $f(r,
\Delta)$ be the smallest integer $k$ such that each $r$-uniform hypergraph of
maximum vertex degree $\Delta$ has a conflict-free coloring with at most $k$
colors. As shown by Tardos and Pach, similarly to a classical Brooks' type
theorem for hypergraphs, $f(r, \Delta)\leq \Delta+1$. Compared to Brooks'
theorem, according to which there is only a couple of graphs/hypergraphs that
attain the $\Delta+1$ bound, we show that there are several infinite classes of
uniform hypergraphs for which the upper bound is attained. We provide bounds on
$f(r, \Delta)$ in terms of~$\Delta$ for large~$\Delta$ and establish the
connection between conflict-free colorings and so-called $\{t, r-t\}$-factors
in $r$-regular graphs. Here, a $\{t, r-t\}$-factor is a factor in which each
degree is either $t$ or $r-t$. Among others, we disprove a conjecture of Akbari
and Kano~[Graphs and Combinatorics 30(4):821--826, 2014] stating that there is
a $\{t,r-t\}$-factor in every $r$-regular graph for odd $r$ and any odd
$t<\frac{r}{3}$.
