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Mixing properties of colourings of the ℤd lattice

Part of: Topological dynamics Graph theory

Published online by Cambridge University Press:  19 October 2020

Noga Alon ,
Raimundo Briceño ,
Nishant Chandgotia ,
Alexander Magazinov  and
Yinon Spinka
Noga Alon
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ08544, USA, and Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv6997801, Israel
Raimundo Briceño
Affiliation:
Pontificia Universidad Católica de Chile, Santiago, Chile
Nishant Chandgotia*
Affiliation:
School of Mathematical Sciences, Hebrew University of Jerusalem, Israel
Alexander Magazinov
Affiliation:
Higher School of Economics, National Research University, 6 Usacheva Street, Moscow119048, Russia
Yinon Spinka
Affiliation:
University of British Columbia, Department of Mathematics, Vancouver, BCV6T 1Z2, Canada
*
*Corresponding author. Email: nishant.chandgotia@gmail.com
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Abstract

We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $q\geq 2d+1$ , the latter holds for any $n \ge 1$ . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 37B10: Symbolic dynamics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 3 , May 2021 , pp. 360 - 373
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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