This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non-perfect colorings.

中文翻译：

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多轨道图案的完美着色

本文研究了具有多个轨道的图案的着色，特别是那些轨道共享颜色的着色。主要问题是确定这种颜色何时变得完美。通过构建完美着色的充分必要条件来表征图案的所有完美着色，可以解决此问题。然后将这些结果应用于对称物体以构建完美和非完美的色彩。