Marthe Bonamy, Paul Ouvrard, Mikaël Rabie, Jukka Suomela, and Jara Uitto
This repository contains code related to the following manuscript: https://arxiv.org/abs/1802.06742
Consider a 2-dimensional torus grid graph that is properly 4-colored. In each step, we can change the color of one node such that the result is again a proper 4-coloring.
Starting from a coloring x, can we reach a coloring y?
Classify the 2x2 tiles of properly 4-colored grids as follows:
Type A (2 tiles):
1 3 2 3
3 2 3 1
Type B (14 tiles):
2 1 4 1
1 4 1 2
3 1 4 1
1 4 1 3
2 3 4 3
3 4 3 2
2 1 4 1
3 4 3 2
2 3 4 3
1 4 1 2
3 2 4 2
1 4 1 3
1 3 2 3
4 2 4 1
No type: everything else.
Lemma: If you change the color of one node in the grid, the total number of type-A tiles mod 2 changes by one if and only if the total number of type-B tiles mod 2 changes by one.
Proof: See the code in this repository.
Corollary: If colorings x and y do not contain any nodes of color 4, and x and y have a different number of type-A tiles mod 2, then it is not possible to reach coloring y from x.
Proof: All type-B tiles contain one node of color 4.
You can verify the lemma by running the Python script check-tiles.py.
It enumerates all 3x3 neighborhoods and all possible ways to change the
color of the middle node, and checks that the claim holds.
It also produces a human-readable output of all cases, available in
the text file check-tiles.txt.