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
EnglishFrançais
Combinatorics, Probability and Computing Combinatorics, Probability and Computing

Article contents

Subcritical Graph Classes Containing All Planar Graphs

Part of: Graph theory Combinatorics

Published online by Cambridge University Press:  22 March 2018

AGELOS GEORGAKOPOULOS  and
STEPHAN WAGNER
AGELOS GEORGAKOPOULOS
Affiliation:
Mathematics Institute, University of Warwick, CV4 7AL, UK
STEPHAN WAGNER
Affiliation:
Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa (e-mail: swagner@sun.ac.za)
Corresponding
E-mail address:
swagner@sun.ac.za
Get access
Rights & Permissions[Opens in a new window]

Abstract

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

MSC classification

Primary: 05C30: Enumeration in graph theory
Secondary: 05A16: Asymptotic enumeration 05C10: Planar graphs; geometric and topological aspects of graph theory 05C80: Random graphs 05C83: Graph minors
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 5 , September 2018 , pp. 763 - 773
Copyright
Copyright © Cambridge University Press 2018 

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.

Footnotes

†

Supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 639046).

‡

Supported by the National Research Foundation of South Africa, grant number 96236.

References

[1] Bernasconi, N., Panagiotou, K. and Steger, A. (2009) The degree sequence of random graphs from subcritical classes. Combin. Probab. Comput. 18 647–681.CrossRefGoogle Scholar
[2] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996) On the Lambert W function. Adv. Comput. Math. 5 329–359.CrossRefGoogle Scholar
[3] Drmota, M. (2009) Random Trees, Springer.CrossRefGoogle Scholar
[4] Drmota, M., Fusy, E., Kang, M., Kraus, V. and Rué, J. (2011) Asymptotic study of subcritical graph classes. SIAM J. Discrete Math. 25 1615–1651.CrossRefGoogle Scholar
[5] Drmota, M. and Noy, M. (2013), Extremal parameters in sub-critical graph classes. In Proceedings of the Tenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), SIAM, pp. 1–7.CrossRefGoogle Scholar
[6] Drmota, M., Ramos, L. and Rué, J. (2017) Subgraph statistics in subcritical graph classes. Random Struct. Alg. 51 631–673.CrossRefGoogle Scholar
[7] Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[8] Georgakopoulos, A. and Wagner, S. (2015) Limits of subcritical random graphs and random graphs with excluded minors. arXiv:1512.03572Google Scholar
[9] Giménez, O. and Noy, M. (2009) Asymptotic enumeration and limit laws of planar graphs. J. Amer. Math. Soc. 22 309–329.CrossRefGoogle Scholar
[10] Goldschmidt, C. (2016) A short introduction to random trees. Mongolian Math. J. 20 53–72.Google Scholar
[11] Harary, F. and Palmer, E. M. (1973) Graphical Enumeration, Academic Press.Google Scholar
[12] Josuat-Vergès, M. (2015) Derivatives of the tree function. Ramanujan J. 38 1–15.CrossRefGoogle Scholar
[13] Kurauskas, V. and McDiarmid, C. (2011) Random graphs with few disjoint cycles. Combin. Probab. Comput. 20 763–775.CrossRefGoogle Scholar
[14] Norine, S., Seymour, P., Thomas, R. and Wollan, P. (2006) Proper minor-closed families are small. J. Combin. Theory Ser. B 96 754–757.CrossRefGoogle Scholar
[15] Noy, M. (2014) Random planar graphs and beyond. In Proceedings of the International Congress of Mathematicians, Seoul 2014, Vol. IV, pp. 407–430.Google Scholar
[16] Panagiotou, K., Stufler, B. and Weller, K. (2016) Scaling limits of random graphs from subcritical classes. Ann. Probab. 44 3291–3334.CrossRefGoogle Scholar
[17] Robertson, N. and Seymour, P. (1986) Graph minors V: Excluding a planar graph. J. Combin. Theory Ser. B 41 92–114.CrossRefGoogle Scholar
[18] Stufler, B. (2015) Random enriched trees with applications to random graphs. arXiv:1504.02006Google 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: 39 *
View data table for this chart

* Views captured on Cambridge Core between 22nd March 2018 - 17th January 2021. This data will be updated every 24 hours.

Hostname: page-component-77fc7d77f9-xz9qf Total loading time: 0.688 Render date: 2021-01-17T23:21:40.058Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags last update: Sun Jan 17 2021 22:54:38 GMT+0000 (Coordinated Universal Time) Feature Flags: { "metrics": true, "metricsAbstractViews": false, "peerReview": true, "crossMark": true, "comments": true, "relatedCommentaries": true, "subject": true, "clr": true, "languageSwitch": true, "figures": false, "newCiteModal": false, "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true }

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.

Subcritical Graph Classes Containing All Planar Graphs
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.

Subcritical Graph Classes Containing All Planar Graphs
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.

Subcritical Graph Classes Containing All Planar Graphs
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *