On random k-out subgraphs of large graphs

We consider random subgraphs of a fixed graph G = (V,E) with large minimum degree. We fix a positive integer k and let Gk be the random subgraph where each v ∈ V independently chooses k random neighbors, making kn edges in all. When the minimum degree δ(G) ≥ ( 12 + ε)n, n = |V | then Gk is k-connected w.h.p. for k = O(1); Hamiltonian for k sufficiently large. When δ(G) ≥ m, then Gk has a cycle of length (1−ε)m for k ≥ kε. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function φ(n) (or φ(m)) where limn→ ∞ φ(n) = 0.