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

TL;DR

This paper proves that determining proper conflict-free and odd k-colorings is NP-complete for bipartite graphs, even planar ones for certain cases, highlighting computational hardness in graph coloring problems.

Contribution

It establishes NP-completeness of proper conflict-free and odd k-coloring problems on bipartite graphs for all k≥3, including the planar case for k=4.

Findings

NP-completeness for all k≥3 on bipartite graphs

NP-completeness of PCF 4-Coloring on planar graphs

Hardness results extend to specific graph classes

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.