Complexity and algorithms for proper conflict-free coloring in graphs

2026-08-11

AI Summary

This study investigates the computational complexity and efficient algorithms for the proper conflict-free (PCF) coloring decision problem on graphs, which requires that every non-isolated vertex has at least one uniquely colored neighbor in its closed neighborhood. Leveraging computational complexity theory and structural graph analysis, the authors establish that PCF $k$-coloring is strongly NP-complete and inapproximable on general graphs, and remains NP-complete even on perfectly orderable bipartite graphs. In contrast, they devise linear-time algorithms for four specific graph classes, including block graphs and proper interval graphs. Furthermore, they prove a universal upper bound $\chi_{\mathrm{pcf}}(G) \leq \omega(G) + 1$, where $\omega(G)$ denotes the clique number of $G$, and characterize the tightness conditions under which this bound is attained.

Abstract

A proper conflict-free (PCF) $k$-coloring of a graph $G$ is a proper $k$-coloring such that there exists a color that appears exactly once in the neighborhood of every non-isolated vertex $v\in V(G)$. The PCF chromatic number, denoted by $χ_{pcf}(G)$, is the least integer $k$ such that there exists a PCF $k$-coloring of $G$. Given a graph $G$ and a positive integer $k$, PCF $k$-\textsc{colorability} is to decide whether $G$ admits a PCF $k$-coloring. Ahn et al. [Discrete Appl. Math. 377 (2025) 10-17] proved that PCF $k$-COLORABILITY is NP-complete for bipartite graphs. We strengthen this result by proving that PCF $k$-COLORABILITY is NP-complete for perfect elimination bipartite graphs, which is a proper subclass of bipartite graphs. We also show that the PCF chromatic number of a graph cannot be approximated within $O(n^{1-\varepsilon})$ unless P=NP, for any $\varepsilon>0$. On the positive side, we provide linear-time algorithms for PCF $k$-$k$-COLORABILITY in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs. We show that $χ_{pcf}(G)\leq ω(G)+1$ for block graphs, proper interval graphs, and pseudo-split graphs (except $C_5$), and we characterize all graphs for which the equality holds.

Problem

• proper conflict-free coloring
• PCF chromatic number
• NP-completeness
• graph coloring
• approximability

Innovation

• proper conflict-free coloring
• NP-completeness
• approximation hardness
• linear-time algorithms
• graph classes