Research supported in part by a USA-Israel BSF grant and by a grant from the Israel Science Foundation.
The phase transition in random graphs: A simple proof
Article first published online: 24 SEP 2012
Copyright © 2012 Wiley Periodicals, Inc.
Random Structures & Algorithms
Volume 43, Issue 2, pages 131–138, September 2013
How to Cite
Krivelevich, M. and Sudakov, B. (2013), The phase transition in random graphs: A simple proof. Random Struct. Alg., 43: 131–138. doi: 10.1002/rsa.20470
Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.
- Issue published online: 24 JUL 2013
- Article first published online: 24 SEP 2012
- Manuscript Accepted: 20 JUN 2012
- Manuscript Received: 18 MAR 2012
- random graphs;
- phase transition;
- giant component;
- long paths
The classical result of Erdős and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around – for any ε > 0 and , all connected components of G(n,p) are typically of size Oε(log n), while for , with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime , the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013