Edge percolation on a random regular graph of low degree

Abstract

Consider a uniformly random regular graph of a fixed degree d ≥ 3, with n vertices. Suppose that each edge is open {closed), with probability p(q = 1 -p), respectively. In 2004 Alon, Benjamini and Stacey proved that p* = (d - 1)-1 is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around p* has width roughly of order n -1/3. More precisely, suppose that p = p(n) is such that ω := n1/3|p - p*| → ∞. If p < p*, then with high probability (whp) the largest component has O((p - p*)-2log n) vertices. If p > p*, and log ω » log log n, then whp the largest component has about n(1 - (pπ + q)d) equivalent to (p - p*) vertices, and the second largest component is of size (p - p*)-2 (log n)1+o(1), at most, where n = (pπ + q)d-1, π ∈(0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains n2/3+o(1) vertices. © Institute of Mathematical Statistics, 2008.