Coloring of the infinite grid

Coloring of the infinite grid

GRAVIER Sylvain

CNRS, University of Grenoble, France sylvain.gravier@ujf-grenoble.fr

VANDOMME Elise

Institut Fourier, University of Grenoble, France e.vandomme@ulg.ac.be

Institute of Mathematics, University of Li`ege, Belgium

Coloring problem

Is it possible to color the infinite grid Z2 with {•, ◦} such that

the coloring satisfies the translations

t = (t, 1)

p = (p, 0)

and, ∀x ∈ Z2, a frame centered on x contains

- α black vertices if its center is black,

- β black vertices if its center is white ?

center

1. Projection

Using the translation t = (t, 1), we can project a frame on the axis {(i, 0) | i ∈ Z}. Let T rans be the set containing a frame and all its translated by multiples of t. We have

Axis t = (t, 1) (0, 0) Axis (0, 0) . . . x₋₁₁ x₋₁₀ x−9 x−8 x−7 x−6 x−5 x−4 x−3 x−2 x−1 x0 x1 x2 x3 x4 x5 x6 x7 . . .

with x_i = #{T ∈ T rans | (i, 0) ∈ T } < ∞ ∀i ∈ Z.

2. Folding

Using the translation p = (p, 0), we can fold an axis of projection on a

cycle _Cp with p weighted vertices. We have

p = (p, 0) . . . x₋₁ x₀ x₁ . . . _xp−1 xp . . . w₂ w1 w₀ w_p−1 w_p−2 with w_i = X k_∈Z x_i+kp < _{∞ ∀i ∈ {0, . . . , p − 1}.}

3. Equivalent problem

Let C_p be the cycle corresponding to the frame and translations given. Is it possible to find a coloring ϕ of C_p,

ϕ : _{{0, . . . , p − 1} → {•, ◦}}

such that, for all rotations ψ of ϕ, we have

- α = P_{i∈ψ−1(•)} w_i if ψ(0) = •, - β = P_{i∈ψ−1(•)} w_i if ψ(0) = ◦ ? 2 w2 1 w1 0 w0 wp−1 p− 1 wp−2 p− 2 Examples : w2 w1 w0 wp₋₁ wp₋₂ monochromatic coloring : ok ! x y x w0 x y x y ok ! x y w0 x y no non-monochromatic solutions if x _{6= y} x y x y x w0 x y x y x t ok !

4. Application to (r, a, b)-codes

An (r, a, b)-code of a graph G = (V, E) is a set of vertices S ⊆ V such that

∀x ∈ S, a = #_{Br(y) | x ∈ B_r(y), y ∈ S}, ∀x ∈ V \ S, b = #_{Br(y) | x ∈ B_r(y), y ∈ S}.

Consider the coloring problem of Z2 with B_r(x), x ∈ Z2, as frames.

Axis t = (2, 1) r = 3 (0, 0) Axis (0, 0) p = (4, 0) . . . _{0 0 1 1 2 3 2 2 3 2 2 3 2 1 1 0 0} . . . 1 + 2 + 3 3 + 2 + 1 2 + 3 + 2 1 + 2 + 2 + 1 6 6 7 6 Axis (0, 0) p = (4, 0) . . . _{0 0 1 1 2 3 2 2 3 2 2 3 2 1 1 0 0} . . . t = (2, 1) p = (4, 0) ⇒ ∃ a (3, 13, 12)-code of Z2 References :

[1] M. A. Axenovich, On multiple coverings of the infinite rectangular grid with balls of constant radius, Discrete Mathematics, 268 (2003), 31–48.

[2] S. A. Puzynina, Perfect colorings of radius r > 1 of the infinite rectangular grid, Siberian Electronic Mathematical Reports, 5 (2008), 283–292.