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2 results

  • Spanning $F$-cycles in random graphs

  • Part of
  • Alberto Espuny Díaz, Yury Person
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 5 / September 2023
    Published online by Cambridge University Press:
    09 June 2023, pp. 833-850
    • Article
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  • We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$-cycles.

  • 8 - Large Subgraphs

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 93-110

  • Summary

    The previous chapter dealt with the existence of small subgraphs of a fixed size. In this chapter, we concern ourselves with the existence of large subgraphs, most notably perfect matchings and Hamilton cycles. Having dealt with perfect matchings, we turn our attention to long paths in sparse random graphs, i.e., in those where we expect a linear number of edges. We next study one of the most celebrated and difficult problems of random graphs: the existence of a Hamilton cycle in a random graph. In the last section of this chapter, we consider the general problem of the existence of arbitrary spanning subgraphs in a random graph