Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-z4vvc Total loading time: 0.25 Render date: 2021-03-04T22:37:07.716Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Combinatorics, Probability and Computing Combinatorics, Probability and Computing

Article contents

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  14 March 2016

MICHAEL KRIVELEVICH ,
MATTHEW KWAN  and
BENNY SUDAKOV
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 6997801, Israel (e-mail: krivelev@post.tau.ac.il)
MATTHEW KWAN
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: matthew.kwan@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: matthew.kwan@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
Corresponding
E-mail address:
krivelev@post.tau.ac.il
matthew.kwan@math.ethz.ch
benjamin.sudakov@math.ethz.ch
Get access
Rights & Permissions[Opens in a new window]

Abstract

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 6 , November 2016 , pp. 909 - 927
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

[1] Bohman, T., Frieze, A., Krivelevich, M. and Martin, R. (2004) Adding random edges to dense graphs. Random Struct. Alg. 24 105–117.CrossRefGoogle Scholar
[2] Bohman, T., Frieze, A. and Martin, R. (2003) How many random edges make a dense graph Hamiltonian? Random Struct. Alg. 22 33–42.CrossRefGoogle Scholar
[3] Bondy, J. (1975) Pancyclic graphs: Recent results. In Infinite and Finite Sets: To Paul Erdős on his 60th Birthday, Vol. 1 (Hajnal, A., Rado, R. and Sós, V. T., eds), Vol. 10 of Colloquia Mathematica Societatis János Bolyai, North-Holland, pp. 181–187.Google Scholar
[4] Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. 3 69–81.CrossRefGoogle Scholar
[5] Dudek, A. and Frieze, A. (2011) Loose Hamilton cycles in random uniform hypergraphs. Electron. J. Combin. 18 #48.Google Scholar
[6] Frieze, A. and Krivelevich, M. (2005) On packing Hamilton cycles in ϵ-regular graphs. J. Combin. Theory Ser. B 94 159–172.CrossRefGoogle Scholar
[7] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Cambridge University Press.CrossRefGoogle Scholar
[8] Johansson, A., Kahn, J. and Vu, V. (2008) Factors in random graphs. Random Struct. Alg. 33 1–28.CrossRefGoogle Scholar
[9] Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations (Miller, R. E. and Thatcher, J. W., eds), Plenum, pp. 85–104.CrossRefGoogle Scholar
[10] Keevash, P., Kühn, D., Mycroft, R. and Osthus, D. (2011) Loose Hamilton cycles in hypergraphs. Discrete Math. 311 544–559.CrossRefGoogle Scholar
[11] Komlós, J. and Simonovits, M. (1996) Szemerédi's Regularity Lemma and its applications in graph theory. In Combinatorics: Paul Erdős is Eighty (Miklós, D., Sós, V., and Szőnyi, T., eds.), Vol. 2 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 295–352.Google Scholar
[12] Korshunov, A. D. (1976) Solution of a problem of Erdős and Rényi on Hamilton cycles in nonoriented graphs. Soviet Math. Doklady 17 760–764.Google Scholar
[13] Krivelevich, M., Sudakov, B. and Tetali, P. (2006) On smoothed analysis in dense graphs and formulas. Random Struct. Alg. 29 180–193.CrossRefGoogle Scholar
[14] Kühn, D., Lapinskas, J., Osthus, D. and Patel, V. (2014) Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments. Proc. London Math. Soc. 109 733–762.CrossRefGoogle Scholar
[15] Łuczak, T., Ruciński, A. and Gruszka, J. (1996) On the evolution of a random tournament. Discrete Math. 148 311–316.CrossRefGoogle Scholar
[16] Moon, J. (1968) Topics on Tournaments, Holt, Rinehart and Winston.Google Scholar
[17] Moon, J. and Moser, L. (1962) Almost all tournaments are irreducible. Canadian Math. Bulletin 5 61–65.CrossRefGoogle Scholar
[18] Pokrovskiy, A. (2014) Edge disjoint Hamiltonian cycles in highly connected tournaments. Int. Math. Res. Not. To appear.Google Scholar
[19] Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.CrossRefGoogle Scholar
[20] Rödl, V. and Ruciński, A. (2010) Dirac-type questions for hypergraphs: A survey (or more problems for Endre to solve). In An Irregular Mind: Szemerédi is 70 (Bárány, I. and Solymosi, J., eds), Vol. 21 of Bolyai Society Mathematical Studies, Springer and János Bolyai Mathematical Society, pp. 561–590.CrossRefGoogle Scholar
[21] Spielman, D. A. and Teng, S.-H. (2003) Smoothed analysis: Motivation and discrete models. In Algorithms and Data Structures: 8th International Workshop, WADS 2003 (Dehne, F., Sack, J. R., and Smid, M., eds), Vol. 2748 of Lecture Notes in Computer Science, Springer, pp. 256–270.CrossRefGoogle Scholar
[22] Spielman, D. A. and Teng, S.-H. (2004) Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. Assoc. Comput. Mach. 51 385–463.CrossRefGoogle Scholar
[23] Tao, T. (2006) Szemerédi's Regularity Lemma revisited. Contrib. Discrete Math. 1 8–28.Google Scholar

Altmetric attention score

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 87 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 4th March 2021. This data will be updated every 24 hours.

15 Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *