  A proper coloring of a graph is \emph{conflict-free} if, for every
non-isolated vertex, some color is used exactly once on its neighborhood. Caro,
Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper
conflict-free coloring with at most $5\Delta(G)/2$ colors and conjectured that
$\Delta(G)+1$ colors suffice for every connected graph $G$ with $\Delta(G)\ge
3$. Our first main result is that even for list-coloring, $\left\lceil
1.6550826\Delta(G)+\sqrt{\Delta(G)}\right\rceil$ colors suffice for every graph
$G$ with $\Delta(G)\ge 10^{8}$; we also prove slightly weaker bounds for all
graphs with $\Delta(G)\ge 750$. These results follow from our more general
framework on proper conflict-free list-coloring of a pair consisting of a graph
$G$ and a "conflict" hypergraph ${\mathcal H}$. As another corollary of our
results in this general framework, every graph has a proper
$(\sqrt{30}+o(1))\Delta(G)^{1.5}$-list-coloring such that every bi-chromatic
component is a path on at most three vertices, where the number of colors is
optimal up to a constant factor. Our proof uses a fairly new type of recursive
counting argument called Rosenfeld counting, which is a variant of the
Lov\'{a}sz Local Lemma or entropy compression.
  We also prove an asymptotically optimal result for a fractional analogue of
our general framework for proper conflict-free coloring for pairs of a graph
and a conflict hypergraph. A corollary states that every graph $G$ has a
fractional $(1+o(1))\Delta(G)$-coloring such that every fractionally
bi-chromatic component has at most two vertices. In particular, it implies that
the fractional analogue of the conjecture of Caro et al.\ holds asymptotically
in a strong sense.
