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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics