TY - JOUR
T1 - Mixing properties of colourings of the ℤdlattice
AU - Alon, Noga
AU - Briceño, Raimundo
AU - Chandgotia, Nishant
AU - Magazinov, Alexander
AU - Spinka, Yinon
N1 - Publisher Copyright:
© 2020 The Author(s). Published by Cambridge University Press.
PY - 2021/5/19
Y1 - 2021/5/19
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.
UR - http://www.scopus.com/inward/record.url?scp=85093690508&partnerID=8YFLogxK
U2 - 10.1017/S0963548320000395
DO - 10.1017/S0963548320000395
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AN - SCOPUS:85093690508
SN - 0963-5483
VL - 30
SP - 360
EP - 373
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 3
ER -