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$G$的适冲突自由色数$\chi_{pcf}(G)$是使得$G$存在一个正常$k$着色，且对每个非孤立顶点，其邻居中恰有一种颜色出现一次的最小整数$k$。图$G$的适奇色数$\chi_{o}(G)$是使得$G$存在一个正常着色，且对每个非孤立顶点，其邻居中存在一种出现奇数次颜色的最小整数$k$。我们称图类$\mathcal{G}$是$\chi_{pcf}$-有界（或$\chi_{o}$-有界）的，若存在函数$f$使得对所有$G \in \mathcal{G}$有$\chi_{pcf}(G) \leq f(\chi(G))$（或$\chi_{o}(G) \leq f(\chi(G))$）。Caro等人（2022）提出了寻找线性$\chi_{pcf}$-有界（或$\chi_{o}$-有界）图类的问题，并以此为基础证明了每个无爪图$G$满足$\chi_{pcf}(G) \le 2\Delta(G)+1$，从而有$\chi_{pcf}(G) \le 4\chi(G)+1$。他们同时猜想所有满足$\Delta(G) \ge 3$的图$G$都有$\chi_{pcf}(G) \le \Delta(G)+1$。本文通过证明无爪图$G$满足$\chi_{pcf}(G) \le \Delta(G)+6$（对于拟线图可改进为$\chi_{pcf}(G) \le \Delta(G)+4$），将该上界改进至接近紧致。进一步地，我们证明凸轮图与置换图是线性$\chi_{pcf}$-有界的。针对后两个结论，我们建立了一个引理：判断遗传图类是否线性$\chi_{pcf}$-有界可归约至判断该类中二部图是否被绝对常数$\chi_{pcf}$-有界。该引理补充了Liu（2022）的一个定理，并激励我们进一步研究二部图中的有界性。此外，我们证明凸二部图不是$\chi_{o}$-有界的，并构造了一类线性$\chi_{o}$-有界但非$\chi_{pcf}$-有界的二部圆图。