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Generalized Majority Colourings of Digraphs

Part of: Graph theory

Published online by Cambridge University Press:  14 August 2017

ANTÓNIO GIRÃO ,
TEERADEJ KITTIPASSORN  and
KAMIL POPIELARZ
ANTÓNIO GIRÃO
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: A.Girao@dpmms.cam.ac.uk)
TEERADEJ KITTIPASSORN
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro, RJ 22451-900, Brazil (e-mail: ping41@mat.puc-rio.br)
KAMIL POPIELARZ
Affiliation:
Department of Mathematics, University of Memphis, Memphis, TN 38152, USA (e-mail: kamil.popielarz@gmail.com)
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Abstract

We almost completely solve a number of problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number k, the smallest number m = m(k) such that every digraph can be coloured with m colours where each vertex has the same colour as at most a 1/k proportion of its out-neighbours. We show that m(k) ∈ {2k − 1,2k}. We also prove a result supporting the conjecture that m(2) = 3. Moreover, we prove similar results for a more general concept called majority choosability.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C20: Directed graphs (digraphs), tournaments
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 6 , November 2017 , pp. 850 - 855
Copyright
Copyright © Cambridge University Press 2017 

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References

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