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On random k-out sub-graphs of large graphs

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $δ(G)\geq (\frac12+ε)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $δ(G) \geq m$, then $G_k$ has a cycle of length $(1-ε)m$ for $k\geq k_ε$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $ϕ(n)$ (or $ϕ(m)$) where $\lim_{n\to\infty}ϕ(n)=0$.