Random Perturbation of Sparse Graphs

In the model of randomly perturbed graphs we consider union a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and binomial random $\mathbb{G}(n,p)$. This was introduced by Bohman, Frieze, Martin for Hamilton cycles their result bridges gap between Dirac's theorem results Pósa Korshunov on threshold in this note extend $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser $\alpha=o(1)$. More precisely, any $\varepsilon>0$ \colon \mathbb{N} \mapsto (0,1)$ show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ fixed constant gives aforementioned if $\alpha=O(1/n)$ part $\mathbb{G}(n,p)$ sufficient cycle. We also discuss embeddings bounded trees other spanning structures model, which lead interesting questions almost into