On the threshold for k-regular subgraphs of random graphs
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
Unable to display preview. Download preview PDF.
- I. Benjamini, G. Kozma and N. Wormald: The mixing time of the giant component of a random graph, Preprint.Google Scholar
- J. Cain and N. Wormald: Encores on cores, Electronic Journal of Combinatorics 13 (2006), RP 81.Google Scholar
- S. Chan and M. Molloy: (k+1)-cores have k-factors, Preprint.Google Scholar
- S. Janson and M. Luczak: A simple solution to the k-core problem, Tech. Report 2005:31, Uppsala.Google Scholar
- J. H. Kim: Poisson cloning model for random graphs, International Congress of Mathematicians, Vol. III, 873–897, Eur. Math. Soc., Zürich, 2006.Google Scholar