Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-q7jt5 Total loading time: 0.292 Render date: 2021-03-03T09:39:14.772Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

A discrepancy version of the Hajnal–Szemerédi theorem

Published online by Cambridge University Press:  30 October 2020

József Balogh
Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, IL 61801, USA, and Moscow Institute of Physics and Technology, Dolgoprudny, Russian Federation
Béla Csaba
Affiliation:
Bolyai Institute, University of Szeged, Hungary
András Pluhár
Affiliation:
Department of Computer Science, University of Szeged, Hungary
Andrew Treglown
Affiliation:
University of Birmingham, Edgbaston, B15 2TT, UK
Corresponding
E-mail address:

Abstract

A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.

Type
Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

Research of this author is partially supported by NSF grants DMS-1500121, DMS-1764123, Arnold O. Beckman Research Award (UIUC) Campus Research Board 18132 and the Langan Scholar Fund (UIUC).

Research of this author was supported in part by the grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary, and by NKFIH grant KH_18 129597.

References

Alexander, J. R., Beck, J. and Chen, W. W. L. (1997) Geometric discrepancy theory and uniform distribution. In Handbook of Discrete and Computational Geometry (Goodman, J. E. et al., eds), pp. 331357. CRC Press.Google Scholar
Alon, N. and Yuster, R. (1996) H-factors in dense graphs. J. Combin. Theory Ser. B 66 269282.CrossRefGoogle Scholar
Balogh, J., Csaba, B., Jing, Y. and Pluhár, A. (2020) On the discrepancies of graphs. Electron. J. Combin. 27 P2.12.CrossRefGoogle Scholar
Beck, J. and Chen, W. W. L. (1987) Irregularities of Distribution, Vol. 89 of Cambridge Tracts in Mathematics. Cambridge University Press.Google Scholar
Bollobás, B. and Eldridge, S. E. (1978) Packing of graphs and applications to computational complexity. J. Combin. Theory Ser. B 25 105124.CrossRefGoogle Scholar
Catlin, P. A. (1976) Embedding subgraphs and coloring graphs under extremal degree conditions. PhD thesis, Ohio State University, Columbus.Google Scholar
Chazelle, B. (2000) The Discrepancy Method. Cambridge University Press.CrossRefGoogle Scholar
Czygrinow, A., DeBiasio, L., Kierstead, H. A. and Molla, T. (2015) An extension of the Hajnal–Szemerédi theorem to directed graphs. Combin. Probab. Comput. 24 754773.CrossRefGoogle Scholar
Czygrinow, A., DeBiasio, L., Molla, T. and Treglown, A. (2018) Tiling directed graphs with tournaments. Forum Math. Sigma 6 e2.CrossRefGoogle Scholar
Erdős, P., Füredi, Z., Loebl, M. and Sós, V. T. (1995) Discrepancy of trees. Stud. Sci. Math. 30 4757.Google Scholar
Gishboliner, L., Krivelevich, M. and Michaeli, P. (2020) Colour-biased Hamilton cycles in random graphs. arXiv:2007.12111Google Scholar
Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications, Vol. II, pp. 601623. János Bolyai Mathematical Society.Google Scholar
Hell, P. and Kirkpatrick, D. G. (1983) On the complexity of general graph factor problems. SIAM J. Comput. 12 601609.Google Scholar
Keevash, P. and Mycroft, R. (2015) A multipartite Hajnal–Szemerédi theorem. J. Combin. Theory Ser. B 114 187236.CrossRefGoogle Scholar
Kierstead, H. A. and Kostochka, A. V. (2008) An Ore-type theorem on equitable coloring. J. Combin. Theory Ser. B 98 226234.CrossRefGoogle Scholar
Komlós, J., Sárközy, G. N. and Szemerédi, E. (1997) Blow-up lemma. Combinatorica 17 109123.CrossRefGoogle Scholar
Komlós, J., Sárközy, G. N. and Szemerédi, E. (1998) Proof of the Seymour conjecture for large graphs. Ann. Combin. 2 4360.CrossRefGoogle Scholar
Komlós, J., Sárközy, G. N. and Szemerédi, E. (2001) Proof of the Alon–Yuster conjecture. Discrete Math. 235 255269.CrossRefGoogle Scholar
Komlós, J. and Simonovits, M. (1996) Szemerédi’s Regularity Lemma and its applications in graph theory. In Combinatorics, Paul Erdős is Eighty, Vol. 2 (Miklós, D., Sós, V. T. and Szőnyi, T., eds), pp. 295352. János Bolyai Mathematical Society.Google Scholar
Kühn, D. and Osthus, D. (2006) Critical chromatic number and the complexity of perfect packings in graphs. In 17th ACM–SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 851859. ACM.CrossRefGoogle Scholar
Kühn, D. and Osthus, D. (2009) The minimum degree threshold for perfect graph packings. Combinatorica 29 65107.CrossRefGoogle Scholar
Sudakov, B. (2017) Robustness of graph properties. In Surveys in Combinatorics 2017, Vol. 440 of London Mathematical Society Lecture Note Series, pp. 372408. Cambridge University Press.CrossRefGoogle Scholar
Szemerédi, E. (1978) Regular partitions of graphs. Problèmes Combinatoires et Théorie des Graphes Colloques Internationaux CNRS 260 399401.Google Scholar
Treglown, A. (2016) A degree sequence Hajnal–Szemerédi theorem. J. Combin. Theory Ser. B 118 1343.CrossRefGoogle Scholar

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: 11 *
View data table for this chart

* Views captured on Cambridge Core between 30th October 2020 - 3rd March 2021. This data will be updated every 24 hours.

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.

A discrepancy version of the Hajnal–Szemerédi theorem
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.

A discrepancy version of the Hajnal–Szemerédi theorem
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.

A discrepancy version of the Hajnal–Szemerédi theorem
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *