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

On the threshold for k-regular subgraphs of random graphs

  • Paweł PrałatEmail author
  • Jacques Verstraëte
  • Nicholas Wormald


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 


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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  • Paweł Prałat
    • 1
    Email author
  • 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

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