The proper conflict-free $k$-coloring problem and the odd $k$-coloring problem are NP-complete on bipartite graphs
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- 10.48550/arXiv.2208.08330
- arXiv:
- arXiv:2208.08330
- Bibcode:
- 2022arXiv220808330A
- Keywords:
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- Computer Science - Computational Complexity;
- Mathematics - Combinatorics;
- 05C15;
- 68Q17
- E-Print:
- 13 pages, 2 figures