K-regular Subgraphs near the K-core Threshold of a Random Graph

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least $e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-\Theta(k)}$.

[1]  Nicholas C. Wormald,et al.  On the threshold for k-regular subgraphs of random graphs , 2007, Comb..

[2]  Nicholas C. Wormald,et al.  Encores on Cores , 2006, Electron. J. Comb..

[3]  N. Wormald,et al.  Models of the , 2010 .

[4]  Joel H. Spencer,et al.  Sudden Emergence of a Giantk-Core in a Random Graph , 1996, J. Comb. Theory, Ser. B.

[5]  W. T. Tutte.  The Factors of Graphs , 1952, Canadian Journal of Mathematics.

[6]  Shoham Letzter,et al.  The property of having a k ‐regular subgraph has a sharp threshold , 2013, Random Struct. Algorithms.

[7]  Bruce A. Reed,et al.  Critical Subgraphs of a Random Graph , 1999, Electron. J. Comb..

[8]  Edward A. Bender,et al.  The asymptotic number of non-negative integer matrices with given row and column sums , 1974, Discret. Math..

[9]  Béla Bollobás,et al.  Random Graphs , 1985 .

[10]  Nicholas C. Wormald,et al.  The mixing time of the giant component of a random graph , 2006, Random Struct. Algorithms.

[11]  Cristiane M. Sato.  On the robustness of random k-cores , 2012, Eur. J. Comb..

[12]  Béla Bollobás,et al.  Regular subgraphs of random graphs , 2006, Random Struct. Algorithms.

[13]  Michael Molloy,et al.  Cores in random hypergraphs and Boolean formulas , 2005, Random Struct. Algorithms.

[14]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[15]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[16]  Michael Molloy,et al.  (k+1)-Cores Have k-Factors , 2012, Combinatorics, Probability and Computing.

[17]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[18]  Pu Gao,et al.  The First k-Regular Subgraph is Large , 2013, Combinatorics, Probability and Computing.

[19]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

[20]  M. Weigt,et al.  Sudden emergence of q-regular subgraphs in random graphs , 2006, cond-mat/0603819.

[21]  N. Wormald.  Models of random regular graphs , 2010 .

[22]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.