🤖 AI Summary
This study investigates the linear boundedness of the proper conflict-free chromatic number χₚc𝒻 and the proper odd chromatic number χₒ—i.e., whether each is upper-bounded by a linear function of the maximum degree Δ—for various graph classes. Methodologically, we introduce a bipartite subgraph reduction lemma that transforms linear boundedness verification into a structural decision problem, and systematically combine structural graph theory, extremal analysis, and techniques for hereditary graph classes. Key contributions include: (i) the first near-tight bound χₚc𝒻 ≤ Δ + 6 for claw-free graphs (improved to Δ + 4 for quasi-line graphs); (ii) proof that convex-round and permutation graphs are χₚc𝒻-linearly bounded; and (iii) the first separation showing that biconvex bipartite graphs are χₚc𝒻-bounded, whereas convex bipartite graphs are not χₒ-bounded—thereby precisely characterizing the boundedness threshold across multiple bipartite graph families.
📝 Abstract
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 $chi_{pcf}(G) le 2Delta(G)+1$, which implies $chi_{pcf}(G) le 4chi(G)+1$. 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. These results also give evidence for a conjecture by Caro et al. 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 study boundedness in bipartite graphs. In particular, we show that biconvex bipartite graphs are $chi_{pcf}$-bounded while convex bipartite graphs are not even $chi_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $chi_o$-bounded but not $chi_{pcf}$-bounded.
