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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  and
GWENAËL JORET
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)
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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.

Keywords

Primary 05C15Secondary 05C10

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 4 , July 2014 , pp. 551 - 570
Copyright
Copyright © The Authors 2014. Published by Cambridge University Press 

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