1. Introduction

A coloring of a plane tessellation (or indeed of any planar graph) is a labeling (by "colors") of the tiles (i.e. regions) of the tessellation in such a way that no two tiles which share an edge have the same color. A coloring is perfect if every isometry of the tessellation permutes the colors. For example, the ordinary black-white coloring of the regular tessellation (4,4,4,4) ("chessboard") is a perfect coloring. If every vertex of the tessellation has the same degree, say n, then a precise coloring is a coloring in which each of the n colors appears at each vertex. The chessboard coloring is perfect, but not precise. Some colorings have the property that every direct isometry permutes the colors, but opposite isometries do not. These colorings are called chirally perfect.

Some of the perfect colorings of the eight Archimedean tessellations of the Euclidean plane are also precise. The purpose of this note is to illustrate all of these precise perfect colorings, as well as those precise colorings which are only chirally perfect.