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## Filters

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Random Matchings in Regular Graphs

1998
*
Combinatorica
*

For a simple d-

doi:10.1007/pl00009817
fatcat:es5pd2s64fbt3he6hzxb3eqdhi
*regular**graph*G, let M be chosen uniformly at*random*from the set of all*matchings*of G, and for x ∈ V (G) let p(x) be the probability that M does not cover x. ... We show that for large d, the p(x)'s and the mean µ and variance σ 2 of |M | are determined to within small tolerances just by d and (*in*the case of µ and σ 2 ) |V (G)|: Theorem For any d-*regular**graph*... We use "*graph*" to mean simple*graph*.)*In*this paper we are concerned with the behavior of M , and*in*particular of the*random*variable ξ = ξ G = |M |, when G is*regular*of large degree. ...##
###
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. ... Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite k-*regular**graphs**in*which the last isolated vertex disappears long before a perfect*matching*appears. ... This is the case, for example,*in**random**regular**graphs*. However,*in*an arbitrary*graph*this need not hold. ...##
###
Randomized Online Matching in Regular Graphs
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

*In*this paper we study the classic online

*matching*problem, introduced

*in*the seminal work of Karp, Vazirani and Vazirani (STOC 1990),

*in*

*regular*

*graphs*. ...

*In*this work, we present a novel

*randomized*algorithm with competitive ratio tending to one on this family of

*graphs*, under adversarial arrival order. ... Online

*Matching*

*in*

*Regular*

*Graphs*.

*In*this work, we study the classic problem of online

*matching*,

*in*the class of d-

*regular*

*graphs*. ...

##
###
The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs
[article]

2021
*
arXiv
*
pre-print

*In*this note, we analyze two

*random*greedy processes on sparse

*random*

*graphs*and hypergraphs with a given degree sequence. ... Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the

*matching*or independent set. ... Indeed, Bohman and Frieze [3] showed that the Karp-Sipser algorithm produces a

*matching*covering all but o(n) vertices

*in*

*random*

*regular*

*graphs*. ...

##
###
Finding perfect matchings in random regular graphs in linear time
[article]

2018
*
arXiv
*
pre-print

We show that the Reduce-Construct algorithm finds a maximum

arXiv:1811.11035v1
fatcat:v6uhqrqtynhuxioh4a4utni7ba
*matching**in**random*k=O(1)-*regular**graphs**in*linear time*in*expectation, as opposed to O(n^3/2) time for the worst-case. ...*In*a seminal paper on finding large*matchings**in*sparse*random**graphs*, Karp and Sipser proposed two algorithms for this task. ...*In*this paper we analyze the Reduce-Construct algorithm for finding perfect*matchings**in**random**regular**graphs*. ...##
###
The number of matchings in random regular graphs and bipartite graphs

1986
*
Journal of combinatorial theory. Series B (Print)
*

Our main aim

doi:10.1016/0095-8956(86)90029-8
fatcat:zxztigkyjzcwzjqvvkknueabpu
*in*this paper is to determine the asymptotic values of the first two moments of the number of l-factors*in**random**regular**graphs*and bipartite*graphs*. ... The systematic study of*random**regular**graphs*was started only recently, when Bender and Canfield [l] determined the asymptotic number of k-*regular**graphs*. ...##
###
Online Matching in Regular Bipartite Graphs with Randomized Adversary
[chapter]

2018
*
Lecture Notes in Computer Science
*

This paper deals with the problem of finding

doi:10.1007/978-3-319-98355-4_17
fatcat:nwymopujkrgyzmi73aaopwotqi
*matchings**in*bipartite*regular**graphs**in*the online model with*randomized*adversary. ... We determine the competitive ratio for 2-*regular**graphs*for any number of*random*decisions made by the adversary. We also give upper and lower bounds for other values of the degree. ... Online*Matching**in**Regular**Graphs*with*Randomized*Adversary*In*the online bipartite*matching*problem, we are given a bipartite*graph*, G = (U ∪ W, E). ...##
###
Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections
[article]

2008
*
arXiv
*
pre-print

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and

arXiv:0807.1277v1
fatcat:ut5xforrerhl5faivaixnv4zfq
*matchings**in*constant degree*regular**graphs*. ... A similar result is established for*matchings*. Our results imply improved bounds for the size of the largest independent set*in*these*graphs*, and provide the first results of this type for*matchings*. ... Acknowledgements The authors would like to thank Alan Frieze and Nick Wormald for a very helpful correspondence explaining the state of the art results*in*the area. ...##
###
Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

2009
*
Combinatorics, probability & computing
*

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and

doi:10.1017/s0963548309990186
fatcat:smhwkggubfhl3h3ctu73hhdz2i
*matchings**in*constant degree*regular**graphs*. ... A similar result is established for*matchings*. Our results imply improved bounds for the size of the largest independent set*in*these*graphs*, and provide the first results of this type for*matchings*. ... Acknowledgements The authors would like to thank Alan Frieze and Nick Wormald for a very helpful correspondence explaining the state of the art results*in*the area. ...##
###
Virtual path network topology optimization using random graphs

1999
*
IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320)
*

This result is made possible by applying the theory of

doi:10.1109/infcom.1999.751382
dblp:conf/infocom/FaragoCB99
fatcat:tlqicvwsknhdtirkoa7rgo5s6y
*random**graphs*to this type of networks. ... Therefore, the paper exemplifies that the theory of*random**graphs*, beyond supporting analysis purposes, may serve as a useful tool*in*the design of algorithms that overcome the "scalability bottleneck, ... Now, taking into account that the complement of a*random*i -1*regular**graph*is a*random*k-*regular**graph*with k = ni , and by ( A ) a*random*k-*regular**graph*almost surely has a perfect*matching*, we have ...##
###
Perfect Matchings in O(n n) Time in Regular Bipartite Graphs
[article]

2010
*
arXiv
*
pre-print

*In*this paper, we give a

*randomized*algorithm that finds a perfect

*matching*

*in*a d-

*regular*

*graph*and runs

*in*O(n n) time (both

*in*expectation and with high probability). ...

*In*

*regular*bipartite

*graphs*, however, a

*matching*is known to be computable

*in*O(m) time (due to Cole, Ost and Schirra). ...

*Matchings*

*in*d-

*Regular*Bipartite

*Graphs*The Basic Algorithm Let G = (P, Q, E) denote the input d-

*regular*

*graph*and let M be a partial

*matching*

*in*G. ...

##
###
Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs

2013
*
SIAM journal on computing (Print)
*

*In*this paper, we give a

*randomized*algorithm that finds a perfect

*matching*

*in*a d-

*regular*

*graph*and runs

*in*O(n log n) time (both

*in*expectation and with high probability). ...

*In*

*regular*bipartite

*graphs*, however, a

*matching*is known to be computable

*in*O(m) time (due to Cole, Ost, and Schirra). ...

*MATCHINGS*

*IN*d-

*REGULAR*BIPARTITE

*GRAPHS*The Basic Algorithm Let G = (P, Q, E) denote the input d-

*regular*

*graph*and let M be a partial

*matching*

*in*G. ...

##
###
Perfect matchings in o(nlogn) time in regular bipartite graphs

2010
*
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
*

*In*this paper, we give a

*randomized*algorithm that finds a perfect

*matching*

*in*a d-

*regular*

*graph*and runs

*in*O(n log n) time (both

*in*expectation and with high probability). ...

*In*

*regular*bipartite

*graphs*, however, a

*matching*is known to be computable

*in*O(m) time (due to Cole, Ost, and Schirra). ...

*MATCHINGS*

*IN*d-

*REGULAR*BIPARTITE

*GRAPHS*The Basic Algorithm Let G = (P, Q, E) denote the input d-

*regular*

*graph*and let M be a partial

*matching*

*in*G. ...

##
###
Page 7571 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

More generally, it is shown here that various ways of building a

*random*r-*regular**graph*as the union of a number of*random**regular**graphs*of smaller degree are equivalent*in*the sense of contiguity. ...*matchings*on 2n vertices and conditioning on the result being an r-*regular**graph*(i.e., conditioning on the event that all*matchings*are dis- joint). ...##
###
On the total domatic number of regular graphs

2012
*
Transactions on Combinatorics
*

We show that the total domatic number of a

doaj:1226c8fd2aa74158906b6c8ef4bf4644
fatcat:2qy3meqhljarjaqsdputqpkwui
*random*r-*regular**graph*is almost surely at most r ... A set S of vertices of a*graph*G = (V;E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex*in*S. ... Thus if an event is almost surely true*in*a*random**graph*constructed from a*random*Hamilton cycle plus a*random**matching*, then it is almost surely true*in*a*random*3-*regular**graph*. ...
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