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For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?

Deryk Osthus*, Hans J�rgen Pr�mel, Anusch Taraz
<span title="2003-01-01">2003</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ujm6quky2va7pmzbitmgsrksbu" style="color: black;">Combinatorica</a> </i> &nbsp;
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-003-0016-1">doi:10.1007/s00493-003-0016-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6ynnot7yivh5pnv7hbqu3jbfre">fatcat:6ynnot7yivh5pnv7hbqu3jbfre</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190222195932/http://pdfs.semanticscholar.org/3c23/1ad0ead246f11f50d2e6e31603d517a09417.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/23/3c231ad0ead246f11f50d2e6e31603d517a09417.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-003-0016-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Spectral gap in random bipartite biregular graphs and applications [article]

Gerandy Brito and Ioana Dumitriu and Kameron Decker Harris
<span title="2021-06-02">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs.  ...  A byproduct of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full-rank with high probability.  ...  We are grateful to an anonymous reviewer for pointing out mistakes in an earlier version of this manuscript, which we have fixed.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1804.07808v6">arXiv:1804.07808v6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zvnporwws5bpln3saiku3b2eoa">fatcat:zvnporwws5bpln3saiku3b2eoa</a> </span>
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Coloring graphs with forbidden bipartite subgraphs [article]

James Anderson, Anton Bernshteyn, Abhishek Dhawan
<span title="2022-01-21">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs F that includes all bipartite graphs.  ...  A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph F, there is a constant c_F > 0 such that if G is an F-free graph of maximum degree Δ, then χ(G) ≤ c_F Δ / logΔ.  ...  Acknowledgements We are grateful to the anonymous referee for helpful suggestions. R e f e r e n c e s  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.05595v3">arXiv:2107.05595v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/djrak6kw4zdxhihmiljdgucc6a">fatcat:djrak6kw4zdxhihmiljdgucc6a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220128043446/https://arxiv.org/pdf/2107.05595v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/12/c5/12c5c27c0eea2f477883a1b1e87ab902d01127f3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.05595v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Density theorems for bipartite graphs and related Ramsey-type results [article]

Jacob Fox, Benny Sudakov
<span title="2007-11-11">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density.  ...  Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers.  ...  We'd like to thank Steve Butler for carefully reading this manuscript. Note added in proof. After this paper was written we learned that D.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0707.4159v2">arXiv:0707.4159v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rpw7shoyjrb3ve3bl7rz7zs6ni">fatcat:rpw7shoyjrb3ve3bl7rz7zs6ni</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0707.4159/0707.4159.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0707.4159v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Spectral gap in random bipartite biregular graphs and applications

Gerandy Brito, Ioana Dumitriu, Kameron Decker Harris
<span title="2021-07-23">2021</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs.  ...  A by-product of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full rank with high probability.  ...  We are grateful to our anonymous reviewers for useful comments and suggestions, and at least in one case for pointing out errors in an earlier version of this manuscript (which have since been fixed).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548321000249">doi:10.1017/s0963548321000249</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nxoihwusrngjbksvpsusxnuamq">fatcat:nxoihwusrngjbksvpsusxnuamq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210726170634/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FF0544B9780460143733B7A5317D6CC1/S0963548321000249a.pdf/div-class-title-spectral-gap-in-random-bipartite-biregular-graphs-and-applications-div.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d1/1b/d11bff20b414dda60a5016c8ab45e4e9826831ce.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548321000249"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> cambridge.org </button> </a>

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

Mathias Schacht, Fabian Schulenburg
<span title="2017-10-16">2017</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wtbq2zeu3fctdhu67pe2vn6hti" style="color: black;">Random structures &amp; algorithms (Print)</a> </i> &nbsp;
For F being a triangle Friedgut, Rödl, Ruciński, and Tetali (2004) established the sharp threshold for the Ramsey property in random graphs.  ...  We obtained a simpler proof of this result which extends to a more general class of graphs F including all cycles.  ...  This leads to the following question: When does the binomial random graph Gpn, pq satisfy A asymptotically almost surely (a.a.s.)?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/rsa.20723">doi:10.1002/rsa.20723</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bkakoecj3ngstpy2ntdr52t2am">fatcat:bkakoecj3ngstpy2ntdr52t2am</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826190747/https://arxiv.org/pdf/1602.02501v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a9/f8/a9f84c3eae5066199f25376d3d25c6d3bb60461b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/rsa.20723"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Correlations in bipartite collaboration networks

Matti Peltomäki, Mikko Alava
<span title="2006-01-20">2006</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/o3oboaa6vjee5ismb4vsvaui2a" style="color: black;">Journal of Statistical Mechanics: Theory and Experiment</a> </i> &nbsp;
Collaboration networks are studied as an example of growing bipartite networks. These have been previously observed to have structure such as positive correlations between nearest-neighbour degrees.  ...  Both of these are analyzed empirically and simulated using various models.  ...  We thank Sergey Dorogovtsev for numerous stimulating and useful discussions, and for a critical reading of an early version of this manuscript.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1742-5468/2006/01/p01010">doi:10.1088/1742-5468/2006/01/p01010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yoghqcorqvee5d3aioxmsqzbvi">fatcat:yoghqcorqvee5d3aioxmsqzbvi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727184124/http://cds.cern.ch/record/865891/files/0508027.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c4/75/c4752cd1e7355fa12bc04310589974289c4a8a61.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1742-5468/2006/01/p01010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> iop.org </button> </a>

A Sequence of Triangle-Free Pseudorandom Graphs

DAVID CONLON
<span title="2016-09-13">2016</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs.  ...  We give an alternative construction for such graphs.  ...  I would like to thank the anonymous referee for a number of helpful remarks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548316000298">doi:10.1017/s0963548316000298</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/stdaj72earhtbbysca22p4wsgu">fatcat:stdaj72earhtbbysca22p4wsgu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200307054729/https://authors.library.caltech.edu/97842/1/1602.03773.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/49/a5/49a5b2a4bcc93f735352ee0ce8d2617510cb3950.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548316000298"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

Extremal subgraphs of random graphs

László Babai, Miklós Simonovits, Joel Spencer
<span title="">1990</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
We shall prove that if L is a 3-chromatic (so called "forbidden") graph, and  ...  Our results also suggest that for random graphs being triangle-free is almost the same as being bipartite, and the same holds for any forbidden 3-chromatic L.  ...  The first construction shows that there are random graphs of specific structure in which the maximum size triangle-free subgraphs and the maximum size bipartite subgraphs are far from each other.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.3190140511">doi:10.1002/jgt.3190140511</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/26cepjl6yrd2vasdarhkf4lswi">fatcat:26cepjl6yrd2vasdarhkf4lswi</a> </span>
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Page 3584 of Mathematical Reviews Vol. , Issue 2004e [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
triangle-free graphs almost surely bipartite?  ...  Dudley Stark (4-LNDQM; London) 2004e:05179 05C80 05A16 05C35 Osthus, Deryk (D-HUMB-II; Berlin) ; Prémel, Hans Jiirgen (D-HUMB-II; Berlin) ; Taraz, Anusch (D-HUMB-II; Berlin) For which densities are random  ... 
<span class="external-identifiers"> </span>
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Asymptotic enumeration, global structure, and constrained evolution

Hans Jürgen Prömel, Angelika Steger, Anusch Taraz
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In this paper, we survey results and methods for such classes of objects. Our focus lies on triangle-free graphs and partially ordered sets.  ...  Consider for example the family of triangle-free graphs. We have already seen that almost all triangle-free graphs are bipartite.  ...  of the random triangle-free graph exhibits two phase transitions with respect to being bipartite: ÿrst it is almost surely bipartite, then it is not, and then it is once again.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00211-9">doi:10.1016/s0012-365x(00)00211-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qnoqatnsobbj7ocfuipfuhorhy">fatcat:qnoqatnsobbj7ocfuipfuhorhy</a> </span>
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The method of hypergraph containers [article]

József Balogh, Robert Morris, Wojciech Samotij
<span title="2018-01-14">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
small family of 'containers' for the independent sets, each of which contains few edges.  ...  This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively  ...  graphs and on extremal and Ramsey properties of random discrete structures inspired and influenced our investigations that led to the hypergraph container theorems.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1801.04584v1">arXiv:1801.04584v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2i524zhma5gbloc7ovoiq2znmi">fatcat:2i524zhma5gbloc7ovoiq2znmi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200924170356/https://arxiv.org/pdf/1801.04584v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a6/0c/a60c06accb33367b7a1ff574e319a626f9aa611f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1801.04584v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Page 4566 of Mathematical Reviews Vol. , Issue 96h [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A variety of techniques are used to show that the size of the random maximal bipartite graph is quadratic in n but of order only n*/? in the triangle-free case.  ...  We address the question of the number of edges of a random maximal P- graph for several properties P, in particular the cases of ‘bipartite’ and ‘triangle-free’.  ... 
<span class="external-identifiers"> </span>
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Page 7624 of Mathematical Reviews Vol. , Issue 2004j [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
set of forbidden substructures, e.g., S graphs which are claw-free, odd-K4-free.  ...  We then use this theorem to show that for any fixed k a random cubic graph contains a K,,-minor asymptotically almost surely.” 2004j:05113 05C85 6swos Bespamyatnikh, Sergei (3-BC-C; Vancouver, BC); Kelarey  ... 
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Recent Developments in Extremal Combinatorics: Ramsey and Turán Type Problems

Benny Sudakov
<span title="">2011</span> <i title="Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India"> Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) </i> &nbsp;
In this paper we discuss some recent advances in this subject, focusing on two topics which played an important role in the development of extremal combinatorics: Ramsey and Turán type questions for graphs  ...  Often such problems are related to other areas including theoretical computer science, geometry, information theory, harmonic analysis and number theory.  ...  Acknowledgments I would like to thank Conlon, Fox, Keevash, Krivelevich and Markman for carefully reading this manuscript and many valuable comments, which greatly improved the presentation.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814324359_0159">doi:10.1142/9789814324359_0159</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6dzzmc7s3bbufe36nr3k7mh5vi">fatcat:6dzzmc7s3bbufe36nr3k7mh5vi</a> </span>
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