On random k‐out subgraphs of large graphs

Abstract

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 v ∈ V independently chooses k random neighbors, making kn edges in all. When the minimum degree δ(G) ≥ ( 1 2 + 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) ≥ m, then G k has a cycle of length (1 − ε)m for k ≥ k ε . By w.h.p. we mean that the probability of nonoccurrence can be bounded by a function φ(n) (or φ(m)) where lim n→∞ φ(n) = 0.

Hamilton Cycles in Random Graphs: a bibliography

“…Frieze and Johansson [96] proved that if H has n vertices and minimum degree at least 1 2 + ε n then H k−out is Hamiltonian w.h.p. for k ≥ k ε .…”

Random k-out Subgraph Leaves only O(n/k) Inter-Component Edges

“…Frieze and Johansson [9] consider random k-out subgraphs of graphs of minimum degree ( 1 2 +ε)n, for some ε > 0. They show that if 2 ≤ k = o( √ log n), then the random k-out subgraph is k-connected with probability tending to 1.…”

“…Thus they generalize the earlier results of Fenner and Frieze [5] for a complete base graph to arbitrary base graphs with sufficiently high minimum degree. Frieze and Johansson [9] points out that the generalization fails for lower degrees: there are connected graphs with minimum degree n/2 where a random k-out subgraph is not even expected to be connected.…”

“…To the best of our knowledge, no result similar to our Corollary 1.6 was known before. It replaces the requirement of a very high minimum degree made in Frieze and Johansson [9] by a much weaker connectivity requirement. However, the resulting random k-out subgraph is only guaranteed to be connected with probability 1/2, not with a probability tending to 1.…”

Connectivity of the k-Out Hypercube

“…Let k ≥ k 1 . If S, L ⊂ V are such that |L| = k − 1 and S ∈ S 1 (L) then the same argument we used to derive (9) implies that k ≤ 3 log 2 |S|. Proof.…”