Boundedness for proper conflict-free and odd colorings

Abstract

The proper conflict-free chromatic number, ${\chi }_{\mathrm{pcf}}\left(G\right)$, of a graph G is the least positive integer 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, ${\chi }_o\left(G\right)$, of G is the least positive integer 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 clearly have $\chi \left(G\right)\le {\chi }_o\left(G\right)\le {\chi }_{\mathrm{pcf}}\left(G\right)$. We say that a graph class $G$ is ${\chi }_{\mathrm{pcf}}$-bounded (${\chi }_o$-bounded) if there is a function f such that ${\chi }_{\mathrm{pcf}}\left(G\right)\le f\left(\chi \right(G\left)\right)$ (${\chi }_o\left(G\right)\le f\left(\chi \right(G\left)\right)$) for every $G\in G$. Caro, Petruševski, and Škrekovski (2023) asked for classes that are linearly ${\chi }_{\mathrm{pcf}}$-bounded (${\chi }_o$-bounded) and, as a starting point, they showed that every claw-free graph G satisfies ${\chi }_{\mathrm{pcf}}\left(G\right)\le 2\Delta \left(G\right)+1$, which implies ${\chi }_{\mathrm{pcf}}\left(G\right)\le 4\chi \left(G\right)+1$.

In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph G satisfies ${\chi }_{\mathrm{pcf}}\left(G\right)\le \Delta \left(G\right)+6$, and even ${\chi }_{\mathrm{pcf}}\left(G\right)\le \Delta \left(G\right)+4$ if it is a quasi-line graph. These results also give further evidence to a conjecture by Caro, Petruševski, and Škrekovski. Moreover, we show that convex-round graphs and permutation graphs are linearly ${\chi }_{\mathrm{pcf}}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly ${\chi }_{\mathrm{pcf}}$-bounded to deciding if the bipartite graphs in the class are ${\chi }_{\mathrm{pcf}}$-bounded by an absolute constant. This lemma complements a theorem of Liu (2024) and motivates us to further study boundedness in bipartite graphs. Among other results, we show that biconvex bipartite graphs are ${\chi }_{\mathrm{pcf}}$-bounded, while convex bipartite graphs are not even ${\chi }_o$-bounded, and we exhibit a class of bipartite circle graphs that is linearly ${\chi }_o$-bounded but not ${\chi }_{\mathrm{pcf}}$-bounded.