On random k-out sub-graphs of large graphs

TL;DR

This paper studies the properties of random sub-graphs formed by each vertex choosing a fixed number of neighbors in large graphs with high minimum degree, showing they are highly connected and contain long cycles.

Contribution

It establishes conditions under which these random sub-graphs are k-connected, Hamiltonian, or contain long cycles, depending on the minimum degree and number of neighbors chosen.

Findings

G_k is k-connected with high probability for large minimum degree and small k.

G_k is Hamiltonian for sufficiently large k.

G_k contains long cycles proportional to the minimum degree.

Abstract

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 $\delta(G)\geq (\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $\phi(n)$ (or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$.