  A vertex coloring of a graph is said to be \textit{conflict-free} with
respect to neighborhoods if for every non-isolated vertex there is a color
appearing exactly once in its (open) neighborhood. As defined in [Fabrici et
al., \textit{Proper Conflict-free and Unique-maximum Colorings of Planar Graphs
with Respect to Neighborhoods}, arXiv preprint], the minimum number of colors
in any such proper coloring of graph $G$ is the PCF chromatic number of $G$,
denoted $\chi_{\mathrm{pcf}}(G)$. In this paper, we determine the value of this
graph parameter for several basic graph classes including trees, cycles,
hypercubes and subdivisions of complete graphs. We also give upper bounds on
$\chi_{\mathrm{pcf}}(G)$ in terms of other graph parameters. In particular, we
show that $\chi_{\mathrm{pcf}}(G) \leq5\Delta(G)/2$ and characterize equality.
Several sufficient conditions for PCF $k$-colorability of graphs are
established for $4\le k\le 6$. The paper concludes with few open problems.
