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Counting maximal near perfect matchings in quasirandom and dense graphs [article]

Yifan Jing, Akbar Rafiey
2019 arXiv   pre-print
In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs.  ...  We provide tight lower and upper bounds on the number of ε-near perfect matchings in generalized quasirandom graphs.  ...  We are also thankful to Heng Guo for pointing out an inaccuracy in the introduction, and bringing reference [26] to our attention after the first version of this paper appeared on arXiv.  ... 
arXiv:1807.04803v3 fatcat:ptxbgs7yzfbhzpid5vme3ex7ze

Almost all Steiner triple systems have perfect matchings [article]

Matthew Kwan
2020 arXiv   pre-print
In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum.  ...  We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class).  ...  of Johansson, Kahn and Vu [24] on perfect matchings in random hypergraphs, and the theory of Keevash and Mycroft [29] characterising when dense hypergraphs have perfect matchings.  ... 
arXiv:1611.02246v5 fatcat:lsge56xdczgm5mzmd4u57dkare

Packing degenerate graphs [article]

Peter Allen, Julia Böttcher, Jan Hladký, Diana Piguet
2017 arXiv   pre-print
Given D and γ>0, whenever c>0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cn n, and total number of edges  ...  at most (1-γ)n2, then G packs into the complete graph K_n.  ...  The contents of this publication reflects only the authors' views and not necessarily the views of the European Commission of the European Union.  ... 
arXiv:1711.04869v1 fatcat:noxlala3fzffreh4ggwni7hpgi

Packing degenerate graphs

Peter Allen, Julia Böttcher, Jan Hladký, Diana Piguet
2019 Advances in Mathematics  
Given D and γ > 0, whenever c > 0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cn log n , and total number  ...  of edges at most (1 − γ) n 2 , then G packs into the complete graph K n .  ...  The contents of this publication reflects only the authors' views and not necessarily the views of the European Commission of the European Union.  ... 
doi:10.1016/j.aim.2019.106739 fatcat:ailkkrqkdjgpfi5gt7mxifvc4i

Sufficient conditions for perfect mixed tilings [article]

Eoin Hurley and Felix Joos and Richard Lang
2022 arXiv   pre-print
Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense.  ...  As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect F-tilings (for an arbitrary fixed graph F) by replacing the F-tiling with the  ...  Acknowledgments We would like to thank Andrew Treglown for stimulating discussions and bringing the work of Ishigami [16, 17] to our attention.  ... 
arXiv:2201.03944v2 fatcat:4s5fwjgth5hytehlncnrj3uuda

Distributed-memory multi-GPU block-sparse tensor contraction for electronic structure

Thomas Herault, Yves Robert, George Bosilca, Robert J. Harrison, Cannada A. Lewis, Edward F. Valeev, Jack J. Dongarra
2021 2021 IEEE International Parallel and Distributed Processing Symposium (IPDPS)  
In this paper, we focus on the critical element of block-sparse tensor algebra, namely binary tensor contraction, and report on an efficient and scalable implementation using the task-focused PaRSEC runtime  ...  Unfortunately, computation with such irregular data structures is a poor match to the dominant imperative, bulk-synchronous parallel programming model.  ...  This leads to the loss of the near-perfect load balance that makes traditional communication-optimal algorithms attain strong scaling.  ... 
doi:10.1109/ipdps49936.2021.00062 fatcat:4d5uwxkmkfdatkhfo76vbgcs5a

Arbeitsgemeinschaft: Limits of Structures

László Lovász, Balázs Szegedy
2013 Oberwolfach Reports  
This approach enables one to study graphs, hypergraphs, permutations, subsets of groups and many other fundamental structures. (2010) : 05Cxx. Mathematics Subject Classification  ...  The goal of the Arbeitsgemeinschaft is to review current progress in the study of very large structures.  ...  The authors further show that φ is continuous, increasing in (p 3 /6, 1/6) and determine a sufficient criterion when the constant function c t := (6t) 1/3 gives the minimizer for (3).  ... 
doi:10.4171/owr/2013/16 fatcat:7nfbg4trsrcvdh35b57wlxuqee

Embedding graphs into larger graphs: results, methods, and problems [article]

Miklós Simonovits, Endre Szemerédi
2019 arXiv   pre-print
Extremal Graph Theory is a very deep and wide area of modern combinatorics.  ...  It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which  ...  We thank to several friends and colleagues the useful discussions about the many topics discussed in this survey.  ... 
arXiv:1912.02068v1 fatcat:2rgpm6wbmvafta74cmcbbxhcs4

The existence of designs via iterative absorption: hypergraph F-designs for arbitrary F [article]

Stefan Glock, Daniela Kühn, Allan Lo, Deryk Osthus
2020 arXiv   pre-print
The graph case r=2 was proved by Wilson in 1975 and forms one of the cornerstones of design theory.  ...  In particular, we obtain a resilience version and a decomposition result for hypergraphs of large minimum degree.  ...  Acknowledgements We are grateful to Ben Barber, John Lapinskas and Richard Montgomery for helpful discussions.  ... 
arXiv:1611.06827v3 fatcat:qdwxebsozvf53kgotvy53ujuka

Substructures in Latin squares [article]

Matthew Kwan, Ashwin Sah, Mehtaab Sawhney, Michael Simkin
2022 arXiv   pre-print
In fact, extending recent work of Kwan, Sah, and Sawhney, we resolve the general large-deviation problem for intercalates in random Latin squares, up to constant factors in the exponent: for any constant  ...  0<δ≤1 we have ℙ[𝐍≤(1-δ)𝔼𝐍]=exp(-Θ(n^2)) and for any constant δ>0 we have ℙ[𝐍≥(1+δ)𝔼𝐍]=exp(-Θ(n^4/3log n)).  ...  Now we use the quasirandomness hypotheses in the lemma statement, which we can apply repeatedly to count copies of any graph H in G j−1,j extending any given set of vertices and edges (we are assuming  ... 
arXiv:2202.05088v4 fatcat:utc5o6cfmfcwnfkeap4pmzwfui

Pseudorandomness

Salil P. Vadhan
2012 Foundations and Trends® in Theoretical Computer Science  
graphs, randomness extractors, list-decodable error-correcting codes, samplers, and pseudorandom generators.  ...  This theory has significance for a number of areas in computer science and mathematics, including computational complexity, algorithms, cryptography, combinatorics, communications, and additive number  ...  Acknowledgments My exploration of pseudorandomness began in my graduate and postdoctoral years at MIT and IAS, under the wonderful guidance of Oded Goldreich, Shafi Goldwasser, Madhu Sudan, and Avi Wigderson  ... 
doi:10.1561/0400000010 fatcat:2xv2ssm7lbhnjktg6l3u5o5kfu

Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading [article]

Zeyu Guo, He Sun
2013 arXiv   pre-print
The runtime of our protocol is tight, and the randomness requirement of Õ( n) random bits almost matches the lower bound of Ω( n) random bits for dense graphs.  ...  We present the first protocol for any expander graph G with n nodes such that, the protocol informs every node in O( n) rounds with high probability, and uses Õ( n) random bits in total.  ...  We would like to thank Luca Trevisan and Avi Wigderson for helpful discussion about our work.  ... 
arXiv:1311.2839v1 fatcat:f3cnyulvzjcqzpt2ttudpjgmku

Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading [chapter]

Zeyu Guo, He Sun
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
The runtime of our protocol is tight, and the randomness requirement of O(log n) random bits almost matches the lower bound of Ω(log n) random bits for dense graphs.  ...  We present the first protocol for any expander graph G with n nodes such that, the protocol informs every node in O(log n) rounds with high probability, and uses O(log n) random bits in total.  ...  We would like to thank Luca Trevisan and Avi Wigderson for helpful discussion about our work.  ... 
doi:10.1137/1.9781611973730.29 dblp:conf/soda/Guo015 fatcat:kkci55pwundg5m3h5a3ctcnp24

Circuit Complexity [chapter]

2011 The Probabilistic Method  
Acknowledgments We are very grateful to all our students and colleagues who contributed to the creation of this second edition by joint research, helpful discussions and useful comments.  ...  Graphs that satisfy any (and thus all) of the properties above are called quasirandom.  ...  Thus either Ô Ö ¾ or the size of the maximum matching in this graph is at least Ô Ö ¾. It follows that our graph contains a set of Ô Ö ¾ edges ½ which form either a star or a matching.  ... 
doi:10.1002/9780470277331.ch12 fatcat:hm7d426ycvezxfnv27o4ertg4e

Extremal hypergraph theory and algorithmic regularity lemma for sparse graphs [article]

Hiêp Hàn, Humboldt-Universität Zu Berlin, Humboldt-Universität Zu Berlin
2017
Moreover, we derive (asymptotically sharp) bounds on minimum degrees of uniform hypergraphs which guarantee the appearance of perfect and nearly perfect matchings.  ...  Roughly speaking the lemma asserts that dense graphs can be approximated by a constant number of bipartite quasi-random graphs, thus, it narrows the gap between deterministic and random graphs.  ...  Perfect and Nearly Perfect Matchings In this chapter, we want to prove the results about perfect and nearly perfect matchings.  ... 
doi:10.18452/16402 fatcat:gvmbuofgbvaitfppvib4jdgqfm
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