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 G(k) be the random subgraph where each vV independently chooses k random neighbors, making kn edges in all. When the minimum degree (G)(12+epsilon)n,n=|V| then G(k) is k-connected w.h.p. for k=O(1); Hamiltonian for k sufficiently large. When (G)m, then G(k) has a cycle of length (1-epsilon)m for kk epsilon. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function phi(n) (or phi(m)) where limn phi(n)=0. (c) 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143-157, 2017