Sudden emergence of a giant k-core in a random graph

The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erdos-Rényi random graph G(n, m) on n vertives, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is, when m is close to n/2. We show that for k ≥ 3, with high probability, a giant k-core appears suddenly when m reaches c_kn/2; here c_k = min_λ>0 λ/π_k(λ) and π_k(λ) = P{Poisson(λ)≥k-1}. In particular, c₃≈3.35. We also demonstrate that, unlike the 2-core, when a k-core appears for the first time it is very likely to be giant, of size ≈p_k(λ_k)n. Here λ_k is the minimum point of λ/π_k(λ) and p_k(λ_k) = P{Poisson(λ_k)≥k}. For k = 3, for instance, the newborn 3-core contains about 0.27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find a k-core if the graph has one.