AI 中文总结
研究无三角形图随机定向中的反馈弧鲁棒性，通过证明特定不等式及概率结果，得出反馈弧比等相关量的渐近性质，证实了Aboulker等人的两个猜想。
AI 中文摘要
对于有向图\(D\)，设\(\vec{\alpha}(D)\)为诱导无环子有向图的最大阶数，\(\vec{\chi}(D)\)为其二色数，\(\mathrm{fas}(D)\)为使\(D\)无环所需删除的最小弧数。我们证明，对于每个固定的\(\zeta\in(0,1/2)\)，存在\(n\)个顶点的无三角形图\(G_n\)，使得均匀随机定向\(D_n\)满足：对于每个大小至少为\(C_\zeta\sqrt{n\log n}\)的顶点集\(U\)，有\((\frac{1}{2}-\zeta)e(G_n[U])\lt\mathrm{fas}(D_n[U])\leq\frac{1}{2}e(G_n[U])\)的概率至少为\(1 - \exp[-\Omega_\zeta(\sqrt{n}(\log n)^{3/2})]\)。上界是通用的，因此反馈弧比可以任意接近最大可能值。特别地，\(\vec{\alpha}(D_n)=O(\sqrt{n\log n})\)，每个线性大小的诱导子有向图的二色数为\(\Omega(\sqrt{n/\log n})\)。这得出\(\vec{\alpha}(n)=\Theta(\sqrt{n\log n})\)和\(\vec{t}(n)=\Theta(\sqrt{\frac{n}{\log n}})\)，其中\(\vec{\alpha}(n)\)和\(\vec{t}(n)\)分别表示所有阶数为\(n\)的无三角形有向图\(D\)中\(\vec{\alpha}(D)\)的最小值和\(\vec{\chi}(D)\)的最大值。这证实了Aboulker、Havet、Pirot和Schabanel的两个猜想。
英文摘要
For an oriented graph $D$, let $\vecα(D)$ be the maximum order of an induced acyclic subdigraph, $\vecχ(D)$ its dichromatic number, and $\mathrm{fas}(D)$ the minimum number of arcs whose deletion makes $D$ acyclic. We prove that for every fixed $ζ\in (0, 1/2)$, there are triangle-free graphs $G_n$ on $n$ vertices such that a uniformly random orientation $D_n$ satisfies, $$ \left( \frac{1}{2} - ζ\right) e(G_n[U]) < \mathrm{fas}(D_n[U]) \leq \frac{1}{2} e(G_n[U]) $$ with probability at least $1-\exp\!\left[-Ω_ζ\!\left(\sqrt n\,(\log n)^{3/2}\right)\right]$ simultaneously for every vertex set $U$ of size at least $C_ζ\sqrt{n\log n}$. The upper bound is universal, so the feedback-arc ratio can be made arbitrarily close to the largest possible value, uniformly over all sufficiently large induced subdigraphs. In particular, $\vecα(D_n) = O(\sqrt{n \log n})$, and every linear-size induced subdigraph has dichromatic number $Ω(\sqrt{n/\log n})$. This yields $\vecα(n) = Θ(\sqrt{n \log n})$ and $\vec{t}(n) = Θ\left(\sqrt{\frac{n}{\log n}}\right)$, where $\vecα(n)$ and $\vec{t}(n)$ denote, respectively, the minimum of $\vecα(D)$ and the maximum of $\vecχ(D)$ over all oriented triangle-free graphs $D$ of order $n$. This confirms two conjectures of Aboulker, Havet, Pirot, and Schabanel.
