# The phase transition in random graphs: A simple proof

@article{Krivelevich2013ThePT, title={The phase transition in random graphs: A simple proof}, author={Michael Krivelevich and Benny Sudakov}, journal={Random Structures \& Algorithms}, year={2013}, volume={43} }

The classical result of Erdős and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1}{n}\end{align*} \end{document} – for any ε > 0 and \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*}p=\frac{1-\epsilon}{n}\end{align*} \end{document}, all connected components of G(n,p) are…

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## References

SHOWING 1-10 OF 20 REFERENCES

### Random graphs

- MathematicsZOR Methods Model. Oper. Res.
- 1989

I study random graphs as a probabilist dealing with some combinatorial structures, and my methods are probabilistic and based on analysis, using for example integration theory, functional analysis, martingales and stochastic integration.

### The Transitive Closure of a Random Digraph

- MathematicsRandom Struct. Algorithms
- 1990

It is shown that, when n is large and np is equal to a constant c greater than 1, it is very likely that all but one of the strong components are very small, and that the unique large strong component contains about Θ2n vertices, where Θ is the unique root in [0, 1] of the equation.

### On the evolution of random graphs

- Mathematics
- 1984

(n) k edges have equal probabilities to be chosen as the next one . We shall 2 study the "evolution" of such a random graph if N is increased . In this investigation we endeavour to find what is the…

### Long cycles in subgraphs of (pseudo)random directed graphs

- MathematicsJ. Graph Theory
- 2012

This work studies the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles by finding a constant c = c(γ) such that the following holds.

### The emergence of a giant component in random subgraphs of pseudo‐random graphs

- MathematicsRandom Struct. Algorithms
- 2004

It is shown that if $\alpha 1$ then G_p contains a unique giant component of size $\Omega(n)$, with all other components of size $O(\log n)$.

### Pseudo-random Graphs

- Computer Science
- 2006

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake,…

### The longest path in a random graph

- MathematicsComb.
- 1981

A random graph with (1+ε)n/2 edges contains a path of lengthcn. A random directed graph with (1+ε)n edges contains a directed path of lengthcn. This settles a conjecture of Erdõs.

### Random graphs

- Mathematics, Computer ScienceSODA '06
- 2006

Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.