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 $\chi_{pcf}(G) \le 4\chi(G)+1$. They also conjectured that any graph $G$ with $\Delta(G) \ge 3$ satisfies $\chi_{pcf}(G) \le \Delta(G)+1$. 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. 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 further study boundedness in bipartite graphs. So among other results, we show that convex bipartite graphs are not $\chi_{o}$-bounded, and a class of bipartite circle graphs that is linearly $\chi_{o}$-bounded but not $\chi_{pcf}$-bounded.

翻译：图$G$的恰当无冲突色数$\chi_{pcf}(G)$是使得$G$存在一个恰当$k$-染色的最小整数$k$，其中对于每个非孤立顶点，其邻居中出现恰好一次的颜色存在。图$G$的恰当奇色数$\chi_{o}(G)$是使得$G$存在一个恰当$k$-染色的最小整数$k$，其中对于每个非孤立顶点，其邻居中出现奇数次数的颜色存在。称图类$\mathcal{G}$是$\chi_{pcf}$-有界（$\chi_{o}$-有界）的，若存在函数$f$使得对于每个$G \in \mathcal{G}$，有$\chi_{pcf}(G) \leq f(\chi(G))$（$\chi_{o}(G) \leq f(\chi(G))$）。Caro等人（2022）提出了关于线性$\chi_{pcf}$-有界（$\chi_{o}$-有界）图类的问题，并作为起点证明了每个无爪图$G$满足$\chi_{pcf}(G) \le 2\Delta(G)+1$，从而推出$\chi_{pcf}(G) \le 4\chi(G)+1$。他们还猜想任何满足$\Delta(G) \ge 3$的图$G$有$\chi_{pcf}(G) \le \Delta(G)+1$。本文中，我们将无爪图的界改进至近乎紧致，证明此类图$G$满足$\chi_{pcf}(G) \le \Delta(G)+6$，且当$G$为准线图时进一步有$\chi_{pcf}(G) \le \Delta(G)+4$。此外，我们证明凸轮图和置换图是线性$\chi_{pcf}$-有界的。对于后两个结论，我们证明了一个引理，它将判定遗传图类是否线性$\chi_{pcf}$-有界的问题简化为判定该类中二分图是否被绝对常数$\chi_{pcf}$-有界的问题。该引理补充了Liu（2022）的一个定理，并促使我们进一步研究二分图的有界性。因此，在其他结果中，我们证明了凸二分图不是$\chi_{o}$-有界的，并构造了一个线性$\chi_{o}$-有界但非$\chi_{pcf}$-有界的二分圆图类。