Skip to main content Accesibility Help
×
×
Home

Polyhedral decompositions of cubic graphs

  • G. Szekeres (a1)
Abstract

A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grünbaum) in terms of polyhedral decompositions alone.

    • 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.

      Polyhedral decompositions of cubic 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.

      Polyhedral decompositions of cubic 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.

      Polyhedral decompositions of cubic graphs
      Available formats
      ×
Copyright
References
Hide All
[1]Berge, Claude, The theory of graphs and its applications (translated by Methuen, Alison Doig, London; John Wiley & Sons, New York; 1962).
[2]Grünbaum, Branko, Conjecture 6, Recent progress in combinatorics, 343 (Proceedings of the Third Waterloo Conference on Combinatorics, May 1968, edited by Tutte, W.T.. Academic Press, New York, London, 1969).
[3]König, D., “Vonalrendszerek kétoldalu felületeken”, Mat. Természettud. Értesitö 29 (1911), 112117.
[4]König, Dénes, Theorie der endlichen und unendlichen Graphen. Kombinatorische Topologie der Streckenkomplexe (Mathematik und ihre Auwendung in Monographien und Lehrbüchern, Band 16. Akad. Verlagsges., Leipzig 1936; reprinted: Chelsea, New York, 1950).
[5]Ore, Oystein, The four-color problem (Pure and Applied Mathematics, 27. Academic Press, New York, London, 1967).
[6]Szekeres, G., “Oriented Tait graphs”, J. Austral. Math. Soc. (to appear).
[7]Tutte, W.T., Connectivity in graphs (University of Toronto Press, Toronto, Ontario; Oxford University Press, London; 1966).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed