5 - Random Graphs
Published online by Cambridge University Press: 15 March 2026
Summary
In this final chapter we study in more detail the properties of the Erdős–Rényi random graph G(n, p). The first half of the chapter introduces the concept of a threshold, covers fundamental results such as the threshold for containing a fixed subgraph and for being connected, and gives Erdős’ classic probabilistic construction of graphs with high girth and chromatic number. The second half of the chapter, which is aimed at Masters students, covers some more advanced material, including the problem of finding spanning subgraphs in G(n, p), the threshold for Hamiltonicity, and the emergence of the giant component. In particular, the final two sections provide striking examples of the power of pseudorandomness.
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- Basic Graph Theory , pp. 174 - 212Publisher: Cambridge University PressPrint publication year: 2026
References
Further reading
Alon, N. and Spencer, J. H., The Probabilistic Method (4th edition), John Wiley & Sons, 2016.Google Scholar
Balogh, J., Morris, R. and Samotij, W., The method of hypergraph containers, Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018.Google Scholar
Bollobás, B. and Riordan, O., Random graphs and branching processes. In: Handbook of Large-scale Random Networks, Springer, 2009.Google Scholar
Böttcher, J., Large-scale structures in random graphs, In: Surveys in Combinatorics, London Mathematical Society Lecture Note Series, Cambridge University Press, 2017.Google Scholar
Caldarelli, G., Scale-Free Networks: Complex Webs in Nature and Technology, Oxford University Press, 2007.CrossRefGoogle Scholar
Frieze, A. and Karoński, M., Introduction to Random Graphs, Cambridge University Press, 2016.Google Scholar
Janson, S., Łuczak, T. and Ruciński, A., Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000.10.1002/9781118032718CrossRefGoogle Scholar
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