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Matchings in regular graphs

D. Naddef, W.R. Pulleyblank
1981 Discrete Mathematics  
We consider k-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.  ... 
doi:10.1016/0012-365x(81)90006-6 fatcat:ld232cnqhbeo5dvzi6ymem6lby

Maximum matchings in regular graphs [article]

Dong Ye
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.  ... 
arXiv:1308.2269v3 fatcat:lod54eloqbfwtio4fcd3ftfkia

Random Matchings in Regular Graphs

Jeff Kahn, Jeong Han Kim
1998 Combinatorica  
For a simple d-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.  ... 
doi:10.1007/pl00009817 fatcat:es5pd2s64fbt3he6hzxb3eqdhi

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

Roman Glebov, Zur Luria, Michael Simkin
2020 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1805.06944v5 fatcat:ras6da3wifeb3lkmmfp7g5ts4a

On maximum matchings in almost regular graphs [article]

Petros A. Petrosyan
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 .  ... 
arXiv:1202.0681v2 fatcat:nfow4e2bazazjg2aun3cmrkmsq

Maximum Matching in Regular and Almost Regular Graphs

Raphael Yuster
2012 Algorithmica  
We present an O(n 2 log n)-time algorithm that finds a maximum 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.  ... 
doi:10.1007/s00453-012-9625-7 fatcat:54h35cildnastmk5jitv2a7haq

Perfect Matchings in $\epsilon$-regular Graphs

Noga Alon, Vojtech Rödl, Andrzej Ruciński
1998 Electronic Journal of Combinatorics  
A super $(d,\epsilon)$-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  ... 
doi:10.37236/1351 fatcat:vdeafxjiwbdovdbrd4fho75qri

Online Matching in Regular Bipartite Graphs

Lali Barrière, Xavier Muñoz, Janosch Fuchs, Walter Unger
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  ... 
doi:10.1142/s0129626418500081 fatcat:h4bak7mntffqvkyfwsocuj3foq

Randomized Online Matching in Regular Graphs [chapter]

Ilan Reuven Cohen, David Wajc
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.  ... 
doi:10.1137/1.9781611975031.62 dblp:conf/soda/CohenW18 fatcat:drza5fz4fzcflhcmragkckfqcm

Matchings in regular graphs from eigenvalues

Sebastian M. Cioabă, David A. Gregory, Willem H. Haemers
2009 Journal of combinatorial theory. Series B (Print)  
Let G be a connected k-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.  ... 
doi:10.1016/j.jctb.2008.06.008 fatcat:yc6i2hkosvdnvbwh55yb4tinni

Perfect matchings in highly cyclically connected regular graphs [article]

Robert Lukoťka, Edita Rollová
2021 arXiv   pre-print
For k≥ 0, let G be a d-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.  ... 
arXiv:1709.08891v2 fatcat:fm2v5f4375hjzdoy5bccpxdmc4

Perfect Matchings in O(n n) Time in Regular Bipartite Graphs [article]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
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.  ... 
arXiv:0909.3346v3 fatcat:rh4z6n5dvzbshmwyhl7scsluaq

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

Shmuel Friedland
2011 arXiv   pre-print
This survey paper deals with upper and lower bounds on the number of k-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.  ... 
arXiv:1112.5632v1 fatcat:us5p5dasbnbwrpsfbcxvbdjcg4

Matching and edge-connectivity in regular graphs

Suil O, Douglas B. West
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.  ... 
doi:10.1016/j.ejc.2010.10.005 fatcat:7lekdqxe5jdqbllf4qbfut7624

Induced Matchings in Regular Graphs and Trees [chapter]

Michele Zito
1999 Lecture Notes in Computer Science  
doi:10.1007/3-540-46784-x_10 fatcat:6gzqwhlqtngvlflvxhmmpstzmy
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