Perfect precise colorings of plane semiregular tilings

TL;DR

This paper investigates perfect precise colorings of plane semiregular tilings, focusing on tilings with valence up to 6, and characterizes their symmetry-preserving colorings.

Contribution

It introduces the concept of perfect precise colorings for semiregular tilings and provides classifications for certain families with valence up to six.

Findings

Characterization of perfect precise colorings for specific semiregular tilings.

Identification of conditions for symmetry-preserving colorings.

Extension of coloring techniques to tilings with valence up to 6.

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 $k\leq 6$.