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Clustered 3-colouring graphs of bounded degree

Part of: Graph theory

Published online by Cambridge University Press:  18 June 2021

Vida Dujmović ,
Louis Esperet ,
Pat Morin ,
Bartosz Walczak  and
David R. Wood
Vida Dujmović
Affiliation:
School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada
Louis Esperet
Affiliation:
Laboratoire G-SCOP (CNRS, Univ. Grenoble Alpes), Grenoble, France
Pat Morin
Affiliation:
School of Computer Science, Carleton University, Ottawa, Canada
Bartosz Walczak
Affiliation:
Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
David R. Wood*
Affiliation:
School of Mathematics, Monash University, Melbourne, Australia
*
*Corresponding author. Email: [email protected]
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Abstract

A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$. The previous best bound was $O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$. The best previous bound for this result was exponential in $\Delta$.

Keywords

clustered colouringplanar graphsgraph minors

MSC classification

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Secondary: 05C15: Coloring of graphs and hypergraphs 05C83: Graph minors
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 1 , January 2022 , pp. 123 - 135
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Research is supported by NSERC.

†

Partially supported by ANR Projects GATO (anr-16-CE40-0009-01) and GrR (anr-18-CE40-0032).

§

Research is supported by NSERC.

Research is partially supported by the National Science Centre of Poland grant 2015/17/D/ST1/00585.

∥

Research is supported by the Australian Research Council.

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