The proper conflict-free chromatic number, $\chi_{pcf}(G)$, of a graph $G$ is the least $k$ such that $G$ has a proper $k$-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The proper odd chromatic number, $\chi_{o}(G)$, of $G$ is the least $k$ such that $G$ has a proper coloring in which for every non-isolated vertex there is a color appearing an odd number of times among its neighbors. We say that a graph class $\mathcal{G}$ is $\chi_{pcf}$-bounded ($\chi_{o}$-bounded) if there is a function $f$ such that $\chi_{pcf}(G) \leq f(\chi(G))$ ($\chi_{o}(G) \leq f(\chi(G))$) for every $G \in \mathcal{G}$. Caro et al. (2022) asked for classes that are linearly $\chi_{pcf}$-bounded ($\chi_{pcf}$-bounded), and as a starting point, they showed that every claw-free graph $G$ satisfies $\chi_{pcf}(G) \le 2\Delta(G)+1$, which implies $\chi_{pcf}(G) \le 4\chi(G)+1$. In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph $G$ satisfies $\chi_{pcf}(G) \le \Delta(G)+6$, and even $\chi_{pcf}(G) \le \Delta(G)+4$ if it is a quasi-line graph. These results also give evidence for a conjecture by Caro et al. Moreover, we show that convex-round graphs and permutation graphs are linearly $\chi_{pcf}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly $\chi_{pcf}$-bounded to deciding if the bipartite graphs in the class are $\chi_{pcf}$-bounded by an absolute constant. This lemma complements a theorem of Liu (2022) and motivates us to study boundedness in bipartite graphs. In particular, we show that biconvex bipartite graphs are $\chi_{pcf}$-bounded while convex bipartite graphs are not even $\chi_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $\chi_o$-bounded but not $\chi_{pcf}$-bounded.

翻译：图$G$的真确无冲突色数$\chi_{pcf}(G)$是满足以下条件的最小整数$k$：$G$存在一个真$k$-染色，使得每个非孤立顶点的邻居中恰好有一种颜色出现一次。图$G$的真确奇色数$\chi_{o}(G)$是满足以下条件的最小整数$k$：$G$存在一个真染色，使得每个非孤立顶点的邻居中至少有一种颜色出现奇数次。称图类$\mathcal{G}$是$\chi_{pcf}$-有界（$\chi_{o}$-有界）的，如果存在函数$f$使得对任意$G \in \mathcal{G}$有$\chi_{pcf}(G) \leq f(\chi(G))$（$\chi_{o}(G) \leq f(\chi(G))$）。Caro等人（2022）提出寻找线性$\chi_{pcf}$-有界（$\chi_{pcf}$-有界）图类的问题，并初步证明每个无爪图$G$满足$\chi_{pcf}(G) \le 2\Delta(G)+1$，从而有$\chi_{pcf}(G) \le 4\chi(G)+1$。本文将该界改进至接近紧确：无爪图$G$满足$\chi_{pcf}(G) \le \Delta(G)+6$，而拟线图更满足$\chi_{pcf}(G) \le \Delta(G)+4$。这些结果也为Caro等人的猜想提供了证据。此外，我们证明凸圆图与置换图是线性$\chi_{pcf}$-有界的。为证明后两个结果，我们建立了一个引理，将判定遗传图类是否线性$\chi_{pcf}$-有界的问题归结为判定该类中二部图是否被绝对常数$\chi_{pcf}$-有界的问题。该引理补充了Liu（2022）的定理，并激励我们研究二部图的有界性。具体而言，我们证明双凸二部图是$\chi_{pcf}$-有界的，而凸二部图甚至不是$\chi_o$-有界的；同时给出一个二部圆图类，它是线性$\chi_o$-有界的但非$\chi_{pcf}$-有界的。