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arXiv · 2607.22044

Feedback-arc robustness in random orientations of pseudorandom triangle-free graphs

Hui Lei·Danning Wang·Yiqiao Wang
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Abstract

For an oriented graph $D$, let $\vec{\alpha}(D)$ be the maximum order of an induced acyclic subdigraph, $\vec{\chi}(D)$ its dichromatic number, and $\mathrm{fas}(D)$ the minimum number of arcs whose deletion makes $D$ acyclic. We prove that for every fixed $\zeta \in (0, 1/2)$, there are triangle-free graphs $G_n$ on $n$ vertices such that a uniformly random orientation $D_n$ satisfies, $$ \left( \frac{1}{2} - \zeta \right) e(G_n[U]) < \mathrm{fas}(D_n[U]) \leq \frac{1}{2} e(G_n[U]) $$ with probability at least $1-\exp\!\left[-\Omega_\zeta\!\left(\sqrt n\,(\log n)^{3/2}\right)\right]$ simultaneously for every vertex set $U$ of size at least $C_\zeta\sqrt{n\log n}$. 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, $\vec{\alpha}(D_n) = O(\sqrt{n \log n})$, and every linear-size induced subdigraph has dichromatic number $\Omega(\sqrt{n/\log n})$. This yields $\vec{\alpha}(n) = \Theta(\sqrt{n \log n})$ and $\vec{t}(n) = \Theta\left(\sqrt{\frac{n}{\log n}}\right)$, where $\vec{\alpha}(n)$ and $\vec{t}(n)$ denote, respectively, the minimum of $\vec{\alpha}(D)$ and the maximum of $\vec{\chi}(D)$ over all oriented triangle-free graphs $D$ of order $n$. This confirms two conjectures of Aboulker, Havet, Pirot, and Schabanel.

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Hui Lei, Danning Wang, Yiqiao Wang. 2026-07-24. Feedback-arc robustness in random orientations of pseudorandom triangle-free graphs. https://arxiv.org/abs/2607.22044

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