The sharp threshold for percolation on expander graphs

Abstract

We consider a random subgraph Gn(p) of a finite graph family Gn= (Vn, En) formed by retaining each edge of Gnindependently with probability p. We show that if Gnis an expander graph with vertices of bounded degree, then for any cn∈ (0, 1) satisfying cn ≫ 1/√ln n and lim sup cn < 1, n→∞ the property that the random subgraph contains a giant component of order cn{pipe}Vn{pipe} has a sharp threshold. © 2013 Versita Warsaw and Springer-Verlag Wien.