  The following relaxation of proper coloring the square of a graph was
recently introduced: for a positive integer $h$, the proper $h$-conflict-free
chromatic number of a graph $G$, denoted $\chi_{pcf}^h(G)$, is the minimum $k$
such that $G$ has a proper $k$-coloring where every vertex $v$ has
$\min\{deg_G(v),h\}$ colors appearing exactly once on its neighborhood. Caro,
Petru\v{s}evski, and \v{S}krekovski put forth a Brooks-type conjecture: if $G$
is a graph with $\Delta(G)\ge 3$, then $\chi_{pcf}^1(G)\leq \Delta(G)+1$. The
best known result regarding the conjecture is $\chi_{pcf}^1(G)\leq
2\Delta(G)+1$, which is implied by a result of Pach and Tardos. We improve upon
the aforementioned result for all $h$, and also enlarge the class of graphs for
which the conjecture is known to be true.
  Our main result is the following: for a graph $G$, if $\Delta(G) \ge h+2$,
then $\chi_{pcf}^h(G)\le (h+1)\Delta(G)-1$; this is tight up to the additive
term as we explicitly construct infinitely many graphs $G$ with
$\chi_{pcf}^h(G)=(h+1)(\Delta(G)-1)$. We also show that the conjecture is true
for chordal graphs, and obtain partial results for quasi-line graphs and
claw-free graphs. Our main result also improves upon a Brooks-type result for
$h$-dynamic coloring.
