TY - JOUR

T1 - Mixing properties of colourings of the Zd lattice

AU - Alon, Noga

AU - Briceño, Raimundo

AU - Chandgotia, Nishant

AU - Magazinov, Alexander

AU - Spinka, Yinon

N1 - Publisher Copyright:
© The Author(s), 2020. Published by Cambridge University Press.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - We study and classify proper q-colourings of the Zd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When q ≦ 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 ≧ d + 2, any proper q-colouring of the boundary of a box of side length n ≧ d + 2 can be extended to a proper q-colouring of the entire box. (3) When q ≧ 2d + 1, the latter holds for any n≧ 1. Consequently, we classify the space of proper q-colourings of the Zd lattice by their mixing properties.

AB - We study and classify proper q-colourings of the Zd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When q ≦ 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 ≧ d + 2, any proper q-colouring of the boundary of a box of side length n ≧ d + 2 can be extended to a proper q-colouring of the entire box. (3) When q ≧ 2d + 1, the latter holds for any n≧ 1. Consequently, we classify the space of proper q-colourings of the Zd lattice by their mixing properties.

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U2 - 10.1017/S0963548320000395

DO - 10.1017/S0963548320000395

M3 - Article

AN - SCOPUS:85093690508

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

ER -