Skip to main content Accessibility help
×
Home

Packing Graphs of Bounded Codegree

Published online by Cambridge University Press:  22 March 2018

WOUTER CAMES VAN BATENBURG
Affiliation:
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands (e-mail: w.camesvanbatenburg@math.ru.nl, ross.kang@gmail.com)
ROSS J. KANG
Affiliation:
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands (e-mail: w.camesvanbatenburg@math.ru.nl, ross.kang@gmail.com)
Corresponding
Rights & Permissions[Opens in a new window]

Abstract

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that if (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumption that G1 or G2 has bounded codegree. In particular, we prove for all t ⩾ 2 that if G1 contains no copy of the complete bipartite graph K2,t and Δ(G1) > 17t · Δ(G2), then (Δ(G1) + 1)(Δ(G2) + 1) ⩽ n + 1 implies that G1 and G2 pack. We also provide a mild improvement if moreover G2 contains no copy of the complete tripartite graph K1,1,s, s ⩾ 1.

Type
Paper
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2018

Footnotes

Supported by NWO grant 613.001.217.

Supported by a NWO Vidi grant (639.032.614).

References

[1] Aigner, M. and Brandt, S. (1993) Embedding arbitrary graphs of maximum degree two. J. London Math. Soc. (2) 48 3951.CrossRefGoogle Scholar
[2] Bollobás, B. and Eldridge, S. E. (1978) Packings of graphs and applications to computational complexity. J. Combin. Theory Ser. B 25 105124.CrossRefGoogle Scholar
[3] Bollobás, B., Janson, S. and Scott, A. (2017) Packing random graphs and hypergraphs. Random Struct. Alg. 51 313.CrossRefGoogle Scholar
[4] Bollobás, B., Kostochka, A. and Nakprasit, K. (2005) On two conjectures on packing of graphs. Combin. Probab. Comput. 14 723736.CrossRefGoogle Scholar
[5] Bollobás, B., Kostochka, A. and Nakprasit, K. (2008) Packing d-degenerate graphs. J. Combin. Theory Ser. B 98 8594.CrossRefGoogle Scholar
[6] Catlin, P. A. (1974) Subgraphs of graphs I. Discrete Math. 10 225233.CrossRefGoogle Scholar
[7] Catlin, P. A. (1976) Embedding subgraphs and coloring graphs under extremal degree conditions. PhD thesis, The Ohio State University. ProQuest LLC, Ann Arbor, MI.Google Scholar
[8] Corrádi, K. (1969) Problem at Schweitzer competition. Mat. Lapok 20 159162.Google Scholar
[9] Csaba, B. (2007) On the Bollobás–Eldridge conjecture for bipartite graphs. Combin. Probab. Comput. 16 661691.CrossRefGoogle Scholar
[10] Csaba, B., Shokoufandeh, A. and Szemerédi, E. (2003) Proof of a conjecture of Bollobás and Eldridge for graphs of maximum degree three. Combinatorica 23 3572.CrossRefGoogle Scholar
[11] Eaton, N. (2000) A near packing of two graphs. J. Combin. Theory Ser. B 80 98103.CrossRefGoogle Scholar
[12] Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications II: (Proc. Colloq., Balatonfüred, 1969), North-Holland, pp. 601–623.Google Scholar
[13] Johansson, A. (1996) Asymptotic choice number for triangle-free graphs. Technical report 91-5, DIMACS.Google Scholar
[14] Kaul, H. and Kostochka, A. (2007) Extremal graphs for a graph packing theorem of Sauer and Spencer. Combin. Probab. Comput. 16 409416.CrossRefGoogle Scholar
[15] Kaul, H., Kostochka, A. and Yu, G. (2008) On a graph packing conjecture by Bollobás, Eldridge and Catlin. Combinatorica 28 469485.CrossRefGoogle Scholar
[16] Molloy, M. and Reed, B. (2002) Graph Colouring and the Probabilistic Method, Vol. 23 of Algorithms and Combinatorics, Springer.CrossRefGoogle Scholar
[17] Sauer, N. and Spencer, J. (1978) Edge disjoint placement of graphs. J. Combin. Theory Ser. B 25 295302.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: 227 *
View data table for this chart

* Views captured on Cambridge Core between 22nd March 2018 - 20th January 2021. This data will be updated every 24 hours.

Access
Open access
Hostname: page-component-76cb886bbf-2rmft Total loading time: 0.212 Render date: 2021-01-20T01:41:06.078Z Query parameters: { "hasAccess": "1", "openAccess": "1", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

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.

Packing Graphs of Bounded Codegree
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.

Packing Graphs of Bounded Codegree
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.

Packing Graphs of Bounded Codegree
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *