Boundedness for proper conflict-free and odd colorings

The proper conflict-free chromatic number, $χ_{pcf}(G)$, 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, $χ_{o}(G)$, 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 $χ_{pcf}$-bounded ($χ_{o}$-bounded) if there is a function $f$ such that $χ_{pcf}(G) \leq f(χ(G))$ ($χ_{o}(G) \leq f(χ(G))$) for every $G \in \mathcal{G}$. Caro et al. (2022) asked for classes that are linearly $χ_{pcf}$-bounded ($χ_{pcf}$-bounded), and as a starting point, they showed that every claw-free graph $G$ satisfies $χ_{pcf}(G) \le 2Δ(G)+1$, which implies $χ_{pcf}(G) \le 4χ(G)+1$. In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph $G$ satisfies $χ_{pcf}(G) \le Δ(G)+6$, and even $χ_{pcf}(G) \le Δ(G)+4$ 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 $χ_{pcf}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly $χ_{pcf}$-bounded to deciding if the bipartite graphs in the class are $χ_{pcf}$-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 $χ_{pcf}$-bounded while convex bipartite graphs are not even $χ_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $χ_o$-bounded but not $χ_{pcf}$-bounded.