The Emergence of a Giant Component in Random Subgraphs of Pseudo-Random Graphs

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