The emergence of a giant component in random subgraphs of pseudo-random graphs

TL;DR

This paper studies the phase transition in the emergence of a giant component in random subgraphs of pseudo-random regular graphs, showing a sharp threshold at a critical probability related to the degree.

Contribution

It establishes the conditions under which a giant component appears in pseudo-random regular graphs, extending classical results to this broader class.

Findings

For <1, maximum component size is logarithmic in n.

For >1, a unique giant component of linear size exists.

The phase transition occurs sharply at =1.

Abstract

Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the adjacency matrix of $G$ is such that the eigenvalue $\lambda$ which is second largest in absolute value satisfies $\lambda=o(d)$. Let $G_p$ with $p=\frac{\alpha}{d}$ be obtained from $G$ by including each edge of $G$ independently with probability $p$. We show that if $\alpha<1$ then whp the maximum component size of $G_p$ is $O(\log n)$ and if $\alpha>1$ then $G_p$ contains a unique giant component of size $\Omega(n)$, with all other components of size $O(\log n)$.