Perfect colorings of the infinite square grid: coverings and twin colors

TL;DR

This paper characterizes all perfect colorings of the infinite square grid, showing they are either orbit colorings or involve twin colors, and provides a classification for the twin color case.

Contribution

It provides a complete characterization of coverings of the infinite square grid by perfect colorings, including a classification of twin color configurations.

Findings

All coverings are either orbit colorings or have twin colors.

Twin colors can be unified without losing perfection, and are classified separately.

The results connect perfect colorings with graph automorphisms and coverings.

Abstract

A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph $G$ is a function $f$ from the set of vertices onto some finite set (of colors) such that every node of color $i$ has exactly $S(i,j)$ neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$ called quotient. If $S$ is an adjacency matrix of some simple graph $T$ on the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors such that unifying them keeps the coloring perfect). The case of twin colors is separately classified. Keywords: perfect coloring, equitable partition, partition design, square grid, rectangular grid, wallpaper group, twin colors, graph covering