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

Boris Pittel, Joel Spencer, Nicholas Wormald

Research output: Contribution to journalArticlepeer-review


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 ckn/2; here ck = minλ>0 λ/πk(λ) and πk(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 ≈pkk)n. Here λk is the minimum point of λ/πk(λ) and pkk) = P{Poisson(λk)≥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.

Original languageEnglish (US)
Pages (from-to)111-151
Number of pages41
JournalJournal of Combinatorial Theory. Series B
Issue number1
StatePublished - May 1996

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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