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arXiv · 2208.08330

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

Jungho Ahn·Seonghyuk Im·Sang-il Oum
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Abstract

A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring} problem asks whether an input graph admits a proper conflict-free $k$-coloring and 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 are NP-complete, even if the input graph is bipartite. Furthermore, we show that the \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph is planar.

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Jungho Ahn, Seonghyuk Im, Sang-il Oum. 2022-08-17. The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs. https://doi.org/10.1016/j.dam.2025.06.026

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