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1 results

  • 5 - Random Graphs

  • Béla Bollobás, University of Cambridge, Robert Morris, IMPA, Rio de Janeiro
  • Book:
    Basic Graph Theory
    Published online:
    15 March 2026
    Print publication:
    26 March 2026, pp 174-212

  • 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.