Random perturbation of sparse graphs

In the model of randomly perturbed graphs we consider the union of a deterministic graph ${\mathcal{G}}_{\alpha }$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pos'{a} and Kor\v{s}unov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $$\mathcal{G}$_\alpha \cup $\mathbb{G}$(n,p)$tosparsergraphswith$\alpha=o(1)$. More precisely, for any $\epsilon >0$ and $\alpha \text{\hspace{0.17em}⁣}:\mathbb{N}\mapsto (0,1)$ we show that a.a.s. ${\mathcal{G}}_{\alpha }\cup \mathbb{G}(n,\beta \mathrm{/}n)$ is Hamiltonian, where $\beta =-(6+\epsilon )\log (\alpha )$. If $\alpha >0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha =O(1\mathrm{/}n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.