Perfect precise colorings of plane semiregular tilings

Abstract

A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a permutation on the finite set of colors. If $\mathcal{T}$ is $k$-valent, then a coloring of $\mathcal{T}$ with $k$ colors is said to be precise if no two tiles of $\mathcal{T}$ sharing the same vertex have the same color. In this work, we obtain perfect precise colorings of some families of $k$-valent semiregular tilings in the plane, where $k\leq 6$.

Figures (18)

• Figure 1: Some semiregular tilings of the plane.
• Figure 2: Perfect precise 3-coloring of some 3-valent semiregular tilings.
• Figure 3: Possible color assignments for a perfect precise 4-coloring of the tiling $(p.q.p.q)$.
• Figure 4: Some perfect precise 4-colorings of the tiling $(p.q.p.q)$.
• Figure 5: Possible color assignments for a perfect precise 4-coloring of the tiling $(p.4.q.4)$.

Theorems & Definitions (15)

• Lemma 1.1
• Lemma 1.2
• Theorem 1.3
• proof
• Lemma 1.4
• proof
• Lemma 1.5
• proof
• Theorem 1.6
• Proposition 2.1