Advertisement

Combinatorica

, Volume 31, Issue 5, pp 565–581 | Cite as

On the threshold for k-regular subgraphs of random graphs

  • Paweł Prałat
  • Jacques Verstraëte
  • Nicholas Wormald
Article

Abstract

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.

Mathematics Subject Classification (2000)

05C80 05C70 05D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Bollobás: Extremal Graph Theory, London Math. Soc. Monographs No. 11, Academic, London (1978).zbMATHGoogle Scholar
  2. [2]
    B. Bollobás, C. Cooper, T. Fenner and A. Frieze: Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k, J. Graph Theory 34(1) (2000), 42–59.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    B. Bollobás, J.H. Kim and J. Verstraëte: Regular subgraphs of random graphs, Random Structures & Algorithms 29 (2006), 1–13.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    I. Benjamini, G. Kozma and N. Wormald: The mixing time of the giant component of a random graph, Preprint.Google Scholar
  5. [5]
    J. Cain and N. Wormald: Encores on cores, Electronic Journal of Combinatorics 13 (2006), RP 81.Google Scholar
  6. [6]
    S. Chan and M. Molloy: (k+1)-cores have k-factors, Preprint.Google Scholar
  7. [7]
    P. Flajolet, D. Knuth and B. Pittel: The first cycles in an evolving graph. Graph theory and combinatorics (Cambridge, 1988). Discrete Math. 75(1–3) (1989), 167–215.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    S. Janson: Cycles and unicyclic components in random graphs, Combin. Probab. Comput. 12(1) (2003), 27–52.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    S. Janson and M. Luczak: A simple solution to the k-core problem, Tech. Report 2005:31, Uppsala.Google Scholar
  10. [10]
    S. Janson, T. Łuczak and A. Ruciński: Random Graphs, Wiley, New York, 2000.CrossRefzbMATHGoogle Scholar
  11. [11]
    J. H. Kim: Poisson cloning model for random graphs, International Congress of Mathematicians, Vol. III, 873–897, Eur. Math. Soc., Zürich, 2006.Google Scholar
  12. [12]
    B. Pittel, J. Spencer and N. Wormald: Sudden emergence of a giant k-core in a random graph, J. Combinatorial Theory, Series B 67 (1996), 111–151.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    M. Pretti and M. Weigt: Sudden emergence of q-regular subgraphs in random graphs, Europhys. Lett. 75, 8 (2006).CrossRefMathSciNetGoogle Scholar
  14. [14]
    W. Tutte: A short proof of the factor theorem for finite graphs, Canadian J. Math. 6 (1954), 347–352.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  • Paweł Prałat
    • 1
  • Jacques Verstraëte
    • 2
  • Nicholas Wormald
    • 3
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  3. 3.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations