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Colouring Planar Graphs With Three Colours and No Large Monochromatic Components

Published online by Cambridge University Press:  01 April 2014

LOUIS ESPERET
Affiliation:
Laboratoire G-SCOP (CNRS, Grenoble-INP), Grenoble, France (e-mail: louis.esperet@g-scop.fr)
GWENAËL JORET
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia (e-mail: gwenael.joret@unimelb.edu.au)

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.

Type
Paper
Copyright
Copyright © The Authors 2014. Published by Cambridge University Press 

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References

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