Skip to main content Accessibility help
×
Home

Colouring Planar Graphs With Three Colours and No Large Monochromatic Components

  • LOUIS ESPERET (a1) and GWENAËL JORET (a2)

Abstract

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic component has at most f(Δ) vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.

    • 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.

      Colouring Planar Graphs With Three Colours and No Large Monochromatic Components
      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.

      Colouring Planar Graphs With Three Colours and No Large Monochromatic Components
      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.

      Colouring Planar Graphs With Three Colours and No Large Monochromatic Components
      Available formats
      ×

Copyright

References

Hide All
[1]Alon, N., Ding, G., Oporowski, B. and Vertigan, D. (2003) Partitioning into graphs with only small components. J. Combin. Theory Ser. B 87 231243.
[2]Berke, R. (2008) Coloring and transversals of graphs. PhD thesis, ETH Zürich. Dissertation 17797.
[3]Grötzsch, H. (1959) Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 8 109120.
[4]Haxell, P., Szabó, T. and Tardos, G. (2003) Bounded size components: Partitions and transversals. J. Combin. Theory Ser. B 88 281297.
[5]Kawarabayashi, K. and Mohar, B. (2007) A relaxed Hadwiger's conjecture for list colorings. J. Combin. Theory Ser. B 97 647651.
[6]Kawarabayashi, K. and Thomassen, C. (2012) From the plane to higher surfaces. J. Combin. Theory Ser. B 102 852868.
[7]Kleinberg, J., Motwani, R., Raghavan, P. and Venkatasubramanian, S. (1997) Storage management for evolving databases. In Proc. 38th Annual IEEE Symposium on Foundations of Computer Science: FOCS 1997, pp. 353–362.
[8]Kostochka, A. (1984) Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica 4 307316.
[9]Linial, N., Matoušek, J., Sheffet, O. and Tardos, G. (2008) Graph colouring with no large monochromatic components. Combin. Probab. Comput. 17 577589.
[10]Linial, N. and Saks, M. (1993) Low diameter graph decompositions. Combinatorica 13 441454.
[11]Mohar, B. and Thomassen, C. (2001) Graphs on Surfaces, The Johns Hopkins University Press.
[12]Thomason, A. (1984) An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95 261265.
[13]Wood, D. R. (2010) Contractibility and the Hadwiger conjecture. Europ. J. Combin. 31 21022109.

Keywords

Colouring Planar Graphs With Three Colours and No Large Monochromatic Components

  • LOUIS ESPERET (a1) and GWENAËL JORET (a2)

Metrics

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