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### Regular subgraphs of random graphs

Béla Bollobás, Jeong Han Kim, Jacques Verstraëte
2006 Random structures & algorithms (Print)
These are the first constant bounds on the average degree in G(n, p) for the existence of a k-regular subgraph. We also discuss the appearance of 3-regular subgraphs in cores of random graphs.  ...  In the case of k = 3, it is also shown that G(n, ρ/n) contains a 3-regular graph with high probability whenever ρ > λ ≈ 5.1494.  ...  Regular Subgraphs of Sparse Random Graphs. In this section, we prove Theorem 1.1.  ...

### Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs [article]

Roman Glebov, Zur Luria, Michael Simkin
2020 arXiv   pre-print
Consider the random process in which the edges of a graph G are added one by one in a random order.  ...  We extend this result to arbitrary k-regular bipartite graphs G on 2n vertices for all k = ω( n/log^1/3 n). Surprisingly, this is not the case for smaller values of k.  ...  We Properties of Random Subgraphs Let k = δn, with δ = ω(log −1/3 n), and fix a k-regular bipartite graph G = (X∪Y, E) on 2n vertices.  ...

### Small subgraphs of random regular graphs

Jeong Han Kim, Benny Sudakov, Van Vu
2007 Discrete Mathematics
Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?  ...  The main aim of this short paper is to answer the following question.  ...  Let 1 d n − 1 be two positive integers, a random regular graph G n,d is obtained by sampling uniformly at random over the set of all simple d-regular graphs on a fixed set of n vertices.  ...

### Distribution of subgraphs of random regular graphs

Zhicheng Gao, N.C. Wormald
2007 Random structures & algorithms (Print)
Let X H be the random variable which is the number of isolated copies of H in a random d-regular graph. Let a denote the order of the automorphism group of H.  ...  Introduction The asymptotic distribution of small subgraphs of a random graph has been basically worked out (see Ruciński  for example).  ...

### Regular induced subgraphs of a random graph [article]

Michael Krivelevich, Benny Sudakov, Nicholas Wormald
2008 arXiv   pre-print
Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n,1/2).  ...  An old problem of Erdős, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices.  ...  Theorem 1.1 Let G be a random graph G(n, 1/2). Then with high probability every induced regular subgraph of G has at most 2n 2/3 vertices.  ...

### Regular induced subgraphs of a random Graph

Michael Krivelevich, Benny Sudakov, Nicholas Wormald
2011 Random structures & algorithms (Print)
Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n, 1/2).  ...  An old problem of Erdős, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices.  ...  We would like to thank the referees of the paper for their careful reading and many helpful remarks.  ...

### Sudden emergence of q-regular subgraphs in random graphs

M Pretti, M Weigt
2006 Europhysics letters
We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large q-regular subgraph, i.e., a subgraph with all vertices having degree equal  ...  For q>3, the q-regular subgraph percolation threshold is found to coincide with that of the q-core.  ...  -In this letter, we have analyzed the emergence (percolation) of q-regular subgraphs in random graphs.  ...

### k-regular subgraphs near the k-core threshold of a random graph [article]

Dieter Mitsche, Michael Molloy, Pawel Pralat
2019 arXiv   pre-print
In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size e^-Θ(k).  ...  We prove that G_n,p=c/n whp has a k-regular subgraph if c is at least e^-Θ(k) above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. an non-empty k-core.  ...  Acknowledgements Parts of this research were conducted while Molloy was an Invited Professor at theÉcole Normale Supérieure, Paris and while Mitsche was visiting Ryerson University.  ...

### On the threshold for k-regular subgraphs of random graphs [article]

Pawel Pralat, Jacques Verstraete, Nicholas Wormald
2007 arXiv   pre-print
Thus the threshold for the appearance of a k-regular subgraph of a random graph is at most the threshold for the (k+2)-core.  ...  We show that for k sufficiently large, the (k + 2)-core of a random graph (n,p) asymptotically almost surely has a spanning k-regular subgraph.  ...  Introduction In this paper, we study the appearance of k-regular subgraphs of random graphs.  ...

### Subgraphs of Random k-Edge-Coloured k-Regular Graphs

PAULETTE LIEBY, BRENDAN D. McKAY, JEANETTE C. McLEOD, IAN M. WANLESS
2009 Combinatorics, probability & computing
Let G = G(n) be a randomly chosen k-edge-coloured k-regular graph with 2n vertices, where k = k(n). Equivalently, G is the union of a random set of k disjoint perfect matchings.  ...  Let h = h(n) be a graph with m = m(n) edges such that m 2 + mk = o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs of G isomorphic to h.  ...  Cycles In this section, we compare some results about cycles in random k-coloured k-regular graphs to those of random k-regular graphs.  ...

### On the threshold for k-regular subgraphs of random graphs

Paweł Prałat, Jacques Verstraëte, Nicholas Wormald
2011 Combinatorica
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 (  ...  The k-core of a graph is the largest subgraph of minimum degree at least k.  ...  Introduction In this paper, we study the appearance of k-regular subgraphs of random graphs.  ...

### Random subgraphs in Cartesian powers of regular graphs

Felix Joos
2012 unpublished
This extends a result of L. Clark, Random subgraphs of certain graph powers, Int. J. Math. Math. Sci., 32(5):285-292, 2002.  ...  Let G be a connected d-regular graph with k vertices.  ...  Introduction For a graph G, we denote by G p a random subgraph of G on the same vertex set which includes every edge of G independently of other edges with probability p ∈ (0, 1).  ...

### The Property of Having a k-Regular Subgraph Has a Sharp Threshold [article]

Shoham Letzter
2013 arXiv   pre-print
We prove that the property of containing a k-regular subgraph in the random graph model G(n,p) has a sharp threshold for k>3.  ...  We also show how to use similar methods to obtain an easy prove for the (known fact of) sharpness of having a non empty k-core for k>3.  ...  subgraph, with probability at least δ/2, the addition of m/2 edges to the graph will yield a graph containing a k-regular subgraph.  ...

### Dense induced subgraphs of dense bipartite graphs [article]

Rose McCarty
2020 arXiv   pre-print
We prove that every bipartite graph of sufficiently large average degree has either a K_t,t-subgraph or an induced subgraph of average degree at least t and girth at least 6.  ...  We conjecture that "6" can be replaced by "k", which strengthens a conjecture of Thomassen. In support of this conjecture, we show that it holds for regular graphs.  ...  Acknowledgement We would like to thank Jacques Verstraete for suggesting Corollary 10, which greatly simplified an earlier version of this paper.  ...

### Page 2412 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews
It investi- gates the distribution of the number of vertices of a given degree in a random subgraph of an arbitrary initial d-regular graph on 7 ver- tices, for suitable functions d = d(n).  ...  McDiarmid (Oxford) 92e:05108 05C80 Palka, Zbigniew (PL-POZN); Ruciriski, Andrzej (PL-POZN) Vertex-degrees in a random subgraph of a regular graph. Studia Sci. Math. Hungar. 25 (1990), no. 3, 209-214.  ...
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