Boundedness for proper conflict-free and odd colorings
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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 𝒢 is χ_pcf-bounded (χ_o-bounded) if there is a function f such that χ_pcf(G) ≤ f(χ(G)) (χ_o(G) ≤ f(χ(G))) for every G ∈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) ≤ 2Δ(G)+1, which implies χ_pcf(G) ≤ 4χ(G)+1. They also conjectured that any graph G with Δ(G) ≥ 3 satisfies χ_pcf(G) ≤Δ(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) ≤Δ(G)+6, and even χ_pcf(G) ≤Δ(G)+4 if it is a quasi-line graph. 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 further study boundedness in bipartite graphs. So among other results, we show that convex bipartite graphs are not χ_o-bounded, and a class of bipartite circle graphs that is linearly χ_o-bounded but not χ_pcf-bounded.
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