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Matchings in regular graphs

1981
*
Discrete Mathematics
*

We consider k-

doi:10.1016/0012-365x(81)90006-6
fatcat:ld232cnqhbeo5dvzi6ymem6lby
*regular**graphs*with specifki edge co~ectivhy and show how some classical theorems and some new results concerning the existence of*matchings*iu such*graphs*can be proved by using the polyhedral ... characterization of Edmonds. ltn addition, we show that iower bounds of LovasZ and Phunmer on the number of perfect ma:l&ings*in*bicritical*graphs*can be improved for cubic bicritical*graphs*. ... Cl*In*the following sectiail we prove several results concerning*matchings**in**regular*igraphs using this technique. ...##
###
Maximum matchings in regular graphs
[article]

2016
*
arXiv
*
pre-print

*In*this note, we confirm the conjecture for all k-

*regular*simple

*graphs*and also k-

*regular*multigraphs with k< 4. ... It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan that every

*graph*G with Δ(G)-δ(G) < 1 has a maximum

*matching*M such that any two M-unsaturated vertices do not share a neighbor. ... Let D be the set of all vertices of a

*graph*G which are not covered by at least one maximum

*matching*, and A, the set of all vertices

*in*V (G) − D adjacent to at least one vertex

*in*D. ...

##
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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

Using a construction due to Goel, Kapralov and Khanna, we show that there exist bipartite k-

arXiv:1805.06944v5
fatcat:ras6da3wifeb3lkmmfp7g5ts4a
*regular**graphs**in*which the last isolated vertex disappears long before a perfect*matching*appears. ... Consider the random process*in*which the edges of a*graph*G are added one by one*in*a random order. ...*In*this paper we consider the threshold p 0 for the appearance of a perfect*matching**in*G(p) where G is a k-*regular*bipartite*graph*on 2n vertices. ...##
###
On maximum matchings in almost regular graphs
[article]

2012
*
arXiv
*
pre-print

*In*this note we show that the conjecture is false for

*graphs*G with Δ(G)-δ(G)=1 and Δ(G)≥ 4, and for r-

*regular*

*graphs*when r≥ 7. ...

*In*the same paper they suggested the following conjecture: every

*graph*G with Δ(G)-δ(G)≤ 1 contains a maximum

*matching*whose unsaturated vertices do not have a common neighbor. ... Next we consider even

*regular*

*graphs*. First we consider the 8-

*regular*

*graph*G shown

*in*Fig. 1 . ...

##
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Maximum Matching in Regular and Almost Regular Graphs

2012
*
Algorithmica
*

We present an O(n 2 log n)-time algorithm that finds a maximum

doi:10.1007/s00453-012-9625-7
fatcat:54h35cildnastmk5jitv2a7haq
*matching**in*a*regular**graph*with n vertices. ... This running time is faster than applying the fastest known general*matching*algorithm that runs*in*O( √ nm)-time for*graphs*with m edges, whenever m = ω(rn 1.5 log n). ... Concluding remarks We presented an O(rn 2 log n) algorithm for maximum*matching**in*r-almost-*regular**graphs*. The most interesting open problem is whether a faster algorithm exists. ...##
###
Perfect Matchings in $\epsilon$-regular Graphs

1998
*
Electronic Journal of Combinatorics
*

A super $(d,\epsilon)$-

doi:10.37236/1351
fatcat:vdeafxjiwbdovdbrd4fho75qri
*regular**graph*on $2n$ vertices is a bipartite*graph*on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$,*in*which the minimum degree and the maximum degree are between ...*in*any such*graph*is at least $(d-2\epsilon)^n n! ... Thus, the number of perfect*matchings**in*any super (d, )-*regular**graph*on 2n vertices is close to the expected number of such*matchings**in*a random bipartite*graph*with edge probability d (which is clearly ...##
###
Online Matching in Regular Bipartite Graphs

2018
*
Parallel Processing Letters
*

*In*this paper we study online bipartite

*matchings*focusing on the particular case of bipartite

*matchings*

*in*

*regular*

*graphs*. ...

*In*an online problem, the input is revealed one piece at a time.

*In*every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. ... Acknowledgements This research is supported

*in*part by the Ministerio de Ciencia e Innovción, Spain, and the European Regional Development Fund under project MTM2014-60127-P, and the Catalan Research Council ...

##
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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*. ... Our main contribution is a novel algorithm which achieves competitive ratio 1 − O √ log d/ √ d

*in*expectation on d-

*regular*

*graphs*. ... Online

*Matching*

*in*

*Regular*

*Graphs*.

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

*matching*,

*in*the class of d-

*regular*

*graphs*. ...

##
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Matchings in regular graphs from eigenvalues

2009
*
Journal of combinatorial theory. Series B (Print)
*

Let G be a connected k-

doi:10.1016/j.jctb.2008.06.008
fatcat:yc6i2hkosvdnvbwh55yb4tinni
*regular**graph*of order n. ... We find a best upper bound (*in*terms of k) on the third largest eigenvalue that is sufficient to guarantee that G has a perfect*matching*when n is even, and a*matching*of order n − 1 when n is odd. ... Acknowledgments The authors are grateful to Mirhamed Shirazi for detecting an error*in*the original proof of Lemma 7 and to the referees for their helpful suggestions. ...##
###
Perfect matchings in highly cyclically connected regular graphs
[article]

2021
*
arXiv
*
pre-print

For k≥ 0, let G be a d-

arXiv:1709.08891v2
fatcat:fm2v5f4375hjzdoy5bccpxdmc4
*regular*cyclically (d-1+2k)-edge-connected*graph*of even order. ... A leaf*matching*operation on a*graph*consists of removing a vertex of degree 1 together with its neighbour from the*graph*. ...*In*this paper, we study perfect*matchings*of highly connected*regular**graphs*. ...##
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Perfect Matchings in O(n n) Time in Regular Bipartite Graphs
[article]

2010
*
arXiv
*
pre-print

*In*

*regular*bipartite

*graphs*, however, a

*matching*is known to be computable

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

*In*this paper we consider the well-studied problem of finding a perfect

*matching*

*in*a d-

*regular*bipartite

*graph*on 2n nodes with m=nd edges. ...

*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. ...

##
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Results and open problems in matchings in regular graphs
[article]

2011
*
arXiv
*
pre-print

This survey paper deals with upper and lower bounds on the number of k-

arXiv:1112.5632v1
fatcat:us5p5dasbnbwrpsfbcxvbdjcg4
*matchings**in**regular**graphs*on N vertices. ... We also consider infinite*regular**graphs*. The analog of k-*matching*is the p-monomer entropy, where p∈ [0,1] is the density of the number of*matchings*. ...*In*§2 we discuss the notions of*matching*polynomials, haffnians and permanents.*In*§3 we give the known upper bounds for*matchings**in**regular**graphs*. ...##
###
Matching and edge-connectivity in regular graphs

2011
*
European journal of combinatorics (Print)
*

*In*this paper, we prove a lower bound for the minimum size of a maximum

*matching*

*in*an l-edge-connected k-

*regular*

*graph*with n vertices, for l ≥ 2 and k ≥ 4. ... Henning and Yeo proved a lower bound for the minimum size of a maximum

*matching*

*in*a connected k-

*regular*

*graphs*with n vertices; it is sharp infinitely often. ... lower bound for the minimum size of a

*matching*

*in*a k-

*regular*l-edge-connected

*graph*with n vertices implies these various results when the parameters are set to appropriate values. ...

##
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Induced Matchings in Regular Graphs and Trees
[chapter]

1999
*
Lecture Notes in Computer Science
*

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