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2 results

  • 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

  • PERFECT 1-FACTORISATIONS OF $K_{16}$

  • Part of
  • MICHAEL J. GILL, IAN M. WANLESS
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 101 / Issue 2 / April 2020
    Published online by Cambridge University Press:
    07 August 2019, pp. 177-185
    Print publication:
    April 2020

  • We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph $K_{16}$. Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of $K_{16}$ with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of $K_{16}$, (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.