Let $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$ -resiliently Hamiltonian. Furthermore, for each $\varepsilon > 0$ , we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$ -resiliently Hamiltonian.This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each $\varepsilon > 0$ , that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$ -resiliently Hamiltonian if $p=\omega(\log^8n/n)$ .

中文翻译：

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随机有向图中的哈密顿性天生具有弹性

让$\{D_M\}_{M\geq 0}$成为n-顶点随机有向图过程，其中$D_0$是空的有向图n顶点和序列中的后续有向图是通过随机均匀地添加新的有向边获得的。对于每个$$\伐普西隆 > 0$$，我们几乎可以肯定地证明，任何有向图$D_M$最小进出度至少为 1 不仅是哈密顿量（如 Frieze 所示），而且在移除边缘时仍然是哈密顿量，只要最多$1/2-\伐普西隆$入射到每个顶点的入边和出边都被删除。我们说这样的有向图是$(1/2-\伐普西隆)$-弹性哈密顿量. 此外，对于每个$\伐普西隆 > 0$，我们证明，几乎可以肯定，每个有向图$D_M$在序列不是$(1/2+\伐普西隆)$- 弹性哈密顿。这改善了 Ferber、Nenadov、Noever、Peter 和 Škorić 的结果$\伐普西隆 > 0$，即二项式随机有向图$D(n,p)$几乎可以肯定$(1/2-\伐普西隆)$- 弹性哈密顿量 if$p=\omega(\log^8n/n)$.