Let $G$ be an $n$-vertex graph, where $\delta (G)\ge \delta n$ for some $\delta :=\delta (n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $\alpha (G)=O\left({\delta }^{2}n\right)$, then perturbing $G$ via the addition of $\omega \left(\frac{\log (1/\delta )}{{\delta }^3}\right)$ random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on $\alpha (G)$ as above and allowing for $\delta =\mathrm{\Omega }(n^{-1/3})$, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. yields a graph containing an almost spanning cycle.

中文翻译：

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随机扰动图中的周期长度

让$G$豆$n$- 顶点图，其中$\delta （G）\ge \delta n$对于一些$\delta :=\delta （n）$。Bohman、Frieze 和 Martin 2003 年的结果断言，如果$\alpha （G）=氧\left({\delta }^{2}n\right)$，然后扰动$G$通过添加$\omega \left(\frac{\mathrm{日志}（1/\delta ）}{{\delta }^3}\right)$随机边，aas 产生哈密顿图。我们证明了上述结果的一些改进和扩展。特别是，保持约束$\alpha （G）$如上所述并允许$\delta =\mathrm{\Omega }（n^{-1/3}）$，我们确定随机边的数量的正确数量级，其添加到$G$aas 产生全环图。此外，我们证明了稀疏图的类似结果，并假设 Chvátal 韧性猜想的正确性，我们处理具有更大独立集的图。最后，在较温和的条件下，我们确定随机边缘数量的正确数量级，这些边缘的添加$G$aas 生成一个包含几乎跨越循环的图。