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Regular subgraphs of random graphs

2006
*
Random structures & algorithms (Print)
*

These are the first constant bounds on the average degree in G(n, p) for the existence

doi:10.1002/rsa.20123
fatcat:dkfbwez5c5adjht2vuxo3hccvu
*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. ...##
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Perfect Matchings in Random Subgraphs of Regular Bipartite Graphs
[article]

2020
*
arXiv
*
pre-print

Consider the

arXiv:1805.06944v5
fatcat:ras6da3wifeb3lkmmfp7g5ts4a
*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. ...##
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Small subgraphs of random regular graphs

2007
*
Discrete Mathematics
*

Given a fixed

doi:10.1016/j.disc.2006.09.032
fatcat:ijo22wyat5b6fmokcsfezweixi
*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. ...##
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Distribution of subgraphs of random regular graphs

2007
*
Random structures & algorithms (Print)
*

Let X H be the

doi:10.1002/rsa.20189
fatcat:i7hw2j2xurfupbpjh2pg24rldy
*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 [5] for example). ...##
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Regular induced subgraphs of a random graph
[article]

2008
*
arXiv
*
pre-print

Motivated by this problem, we consider the order

arXiv:0808.2023v1
fatcat:culawueqdrhhzng2ceom5mzbmy
*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. ...##
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Regular induced subgraphs of a random Graph

2011
*
Random structures & algorithms (Print)
*

Motivated by this problem, we consider the order

doi:10.1002/rsa.20324
fatcat:pdu2no663bfcrdjheuur6aqipe
*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. ...##
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Sudden emergence of q-regular subgraphs in random graphs

2006
*
Europhysics letters
*

We investigate the computationally hard problem whether a

doi:10.1209/epl/i2006-10070-4
fatcat:komqtzwvnrhuxjhyxnmrygmnl4
*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*. ...##
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k-regular subgraphs near the k-core threshold of a random graph
[article]

2019
*
arXiv
*
pre-print

In particular, this pins down the threshold for the appearance

arXiv:1804.04173v2
fatcat:3aan6rp3cbcxxcuth5ls4u27oq
*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. ...##
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On the threshold for k-regular subgraphs of random graphs
[article]

2007
*
arXiv
*
pre-print

Thus the threshold for the appearance

arXiv:0706.1103v1
fatcat:vcf5dyl3brfrrosc4eabl2bkgi
*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*. ...##
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Subgraphs of Random k-Edge-Coloured k-Regular Graphs

2009
*
Combinatorics, probability & computing
*

Let G = G(n) be a randomly chosen k-edge-coloured k-

doi:10.1017/s0963548309009882
fatcat:fp73ycohmfay5gz3cxdkl4zaha
*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*. ...##
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On the threshold for k-regular subgraphs of random graphs

2011
*
Combinatorica
*

We show that for k sufficiently large, the threshold for the appearance

doi:10.1007/s00493-011-2545-3
fatcat:uosvjifcp5gyhkgga4mucnbszq
*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*. ...##
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Random subgraphs in Cartesian powers of regular graphs

2012
unpublished

This extends a result

fatcat:5hcwvzn4incg5jn3k5dx7cflce
*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). ...##
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The Property of Having a k-Regular Subgraph Has a Sharp Threshold
[article]

2013
*
arXiv
*
pre-print

We prove that the property

arXiv:1310.5141v1
fatcat:xqajf6wnercgheaoz4rgpwn7l4
*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*. ...##
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Dense induced subgraphs of dense bipartite graphs
[article]

2020
*
arXiv
*
pre-print

We prove that every bipartite

arXiv:2004.00035v1
fatcat:hkcqbln4rnfb3gkjyqw7unj67i
*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. ...##
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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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