AI 中文总结
该研究证明完美消去二分图的PCF k着色问题为NP完全，PCF色数无法在O(n^(1-ε))内逼近，还为四类图的PCF k着色提供线性时间算法并确定其色数上界。
AI 中文摘要
图G的真无冲突（PCF）k着色是一种真k着色，使得对G的每个非孤立顶点v∈V(G)的邻域中，存在一种颜色恰好出现一次。PCF色数记为χ_pcf(G)，是存在PCF k着色的最小整数k。给定图G和正整数k，PCF k着色问题是判定G是否存在PCF k着色。Ahn等人[《离散应用数学》377卷（2025）10-17页]证明了二分图的PCF k着色问题是NP完全的。我们通过证明完美消去二分图（二分图的真子类）的PCF k着色问题是NP完全的，强化了该结果。我们还证明，除非P=NP，对于任意ε>0，PCF色数无法在O(n^(1-ε))的近似因子内逼近。在正面结果方面，我们为块图、真区间图、链图和伪分裂图的PCF k着色问题提供了线性时间算法。我们证明，对于块图、真区间图和伪分裂图（除C₅外），χ_pcf(G)≤ω(G)+1，并刻画了所有满足该等式的图。
英文摘要
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 $k$-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 $k$-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.
