Abstract

Let G be a d-regular graph of growing degree on n vertices, and form a random subgraph G_p of G by retaining edge of G independently with probability p=p(d). Which conditions on G suffice to observe a phase transition at p=1/d, similar to that in the binomial random graph G(n,p), or, say, in a random subgraph of the binary hypercube Q^d?

We argue that in the supercritical regime p=(1+epsilon)/d, epsilon>0 being a small constant, postulating that every vertex subset S of G of at most n/2 vertices has its edge boundary at least C|S|, for some large enough constant C=C(\epsilon)>0, suffices to guarantee the likely appearance of the giant component in G_p. Moreover, its asymptotic order is equal to that in the random graph G(n,(1+\epsilon)/n), and all other components are typically much smaller.

We further give examples demonstrating the tightness of this result in several key senses.

This is joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.