Remarks on proper conflict-free colorings of graphs

A vertex coloring of a graph is said to be conflict-free with respect neighborhoods if for every non-isolated there color appearing exactly once in its (open) neighborhood. As defined [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings Planar Graphs Respect Neighborhoods , arXiv preprint], the minimum number colors any such proper G PCF chromatic denoted ? pcf ( ) . In this paper, we determine value parameter several basic classes including trees, cycles, hypercubes subdivisions complete graphs. We also give upper bounds on terms other parameters. particular, show that ? 5 ? / 2 characterize equality. Several sufficient conditions k -colorability graphs are established 4 6 The paper concludes few open problems.