Boundedness for proper conflict-free and odd colorings
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2023
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| _version_ | 1866912783646326784 |
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| author | Jiménez, Andrea Knauer, Kolja Lintzmayer, Carla Negri Matamala, Martín Peña, Juan Pablo Quiroz, Daniel A. Sambinelli, Maycon Wakabayashi, Yoshiko Yu, Weiqiang Zamora, José |
| author_facet | Jiménez, Andrea Knauer, Kolja Lintzmayer, Carla Negri Matamala, Martín Peña, Juan Pablo Quiroz, Daniel A. Sambinelli, Maycon Wakabayashi, Yoshiko Yu, Weiqiang Zamora, José |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_00170 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundedness for proper conflict-free and odd colorings Jiménez, Andrea Knauer, Kolja Lintzmayer, Carla Negri Matamala, Martín Peña, Juan Pablo Quiroz, Daniel A. Sambinelli, Maycon Wakabayashi, Yoshiko Yu, Weiqiang Zamora, José Combinatorics Discrete Mathematics 05C15, 05C62 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. |
| title | Boundedness for proper conflict-free and odd colorings |
| topic | Combinatorics Discrete Mathematics 05C15, 05C62 |
| url | https://arxiv.org/abs/2308.00170 |