Boundedness for proper conflict-free and odd colorings

TL;DR

The paper advances our understanding of boundedness for two coloring notions—proper conflict-free colorings and proper odd colorings—across several graph classes. It significantly tightens upper bounds for claw-free and quasi-line graphs, showing $\chi_{\mathrm{pcf}}(G) \le \Delta(G)+6$ for claw-free and $\le \Delta(G)+4$ for quasi-line graphs, and demonstrates linear $\chi_{\mathrm{pcf}}$-boundedness for convex-round and permutation graphs. A unifying framework reduces linear boundedness questions to the bipartite case via a key lemma, and the subdivision results relate $\chi(G)$ to $\chi_{\mathrm{pcf}}(S(G))$ and $\chi_{\mathrm{o}}(S(G))$ when $\chi(G)$ is small. The paper also presents stark separations between $\chi_{\mathrm{o}}$ and $\chi_{\mathrm{pcf}}$ in bipartite circle graphs and convex bipartite graphs, highlighting the nuanced role bipartite structure plays in these boundedness questions. Collectively, these results contribute constructive bounds, a clear reduction pathway, and insight into the limits of linear boundedness for these coloring variants.

Abstract

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.

Figures (1)

• Figure 1: The interval representation for $H_4$ and its representations as chords in a circle.

Theorems & Definitions (52)

• Theorem 1.2: Liu Liu2024
• Lemma 1.2
• Theorem 1.2
• Theorem 1.2
• Lemma 1.2
• Theorem 1.3
• Proposition 1.3
• Theorem 1.3
• Theorem 1.3
• Lemma 2.1