Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order
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arXiv
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| Format: | Preprint |
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2024
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| author | Aboulker, Pierre Havet, Frédéric Pirot, François Schabanel, Juliette |
| author_facet | Aboulker, Pierre Havet, Frédéric Pirot, François Schabanel, Juliette |
| contents | Let $D$ be a digraph. Its acyclic number $\vecα(D)$ is the maximum order of an acyclic induced subdigraph and its dichromatic number $\vecχ(D)$ is the least integer $k$ such that $V(D)$ can be partitioned into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec t(n)$ which are the minimum of $\vecα(D)$ and the maximum of $\vecχ(D)$, respectively, over all oriented triangle-free graphs of order $n$. For every $ε>0$ and $n$ large enough, we show $(1/\sqrt{2} - ε) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + ε) \sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order Aboulker, Pierre Havet, Frédéric Pirot, François Schabanel, Juliette Combinatorics Discrete Mathematics 05C20, 05C55 Let $D$ be a digraph. Its acyclic number $\vecα(D)$ is the maximum order of an acyclic induced subdigraph and its dichromatic number $\vecχ(D)$ is the least integer $k$ such that $V(D)$ can be partitioned into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec t(n)$ which are the minimum of $\vecα(D)$ and the maximum of $\vecχ(D)$, respectively, over all oriented triangle-free graphs of order $n$. For every $ε>0$ and $n$ large enough, we show $(1/\sqrt{2} - ε) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and $\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + ε) \sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25 vertices with dichromatic number~3, and show that every oriented triangle-free graph of order at most 17 has dichromatic number at most 2. |
| title | Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order |
| topic | Combinatorics Discrete Mathematics 05C20, 05C55 |
| url | https://arxiv.org/abs/2403.02298 |