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A proper coloring of a graph is \emph{proper conflict-free} if everynon-isolated vertex $v$ has a neighbor whose color is unique in theneighborhood of $v$. A proper coloring of a graph is \emph{odd} if for everynon-isolated vertex $v$, there is a color appearing an odd number of times inthe neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring}problem asks whether an input graph admits a proper conflict-free $k$-coloringand the \textsc{Odd $k$-Coloring} asks whether an input graph admits an odd$k$-coloring. We show that for every integer $k\geq3$, both problems areNP-complete, even if the input graph is bipartite. Furthermore, we show thatthe \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph isplanar.

The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs