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

Authors:
Alan Frieze;Michael Krivelevich;Ryan Martin
Affiliations:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania;Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
Venue:
Random Structures & Algorithms
Year:
2004

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Abstract

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 Gp with p = α/d be obtained from G by including each edge of G independently with probability p. We show that if α whp the maximum component size of Gp is O(log n) and if α 1, then Gp contains a unique giant component of size Ω(n), with all other components of size O(log n).