Abila Agatha Kristel, De Las Peñas Ma Louise Antonette, Tomenes Mark
Department of Mathematics, Ateneo de Manila University, Loyola Heights, Quezon City, Metro Manila, 1108, Philippines.
Acta Crystallogr A Found Adv. 2024 Sep 1;80(Pt 5):367-378. doi: 10.1107/S2053273324005643. Epub 2024 Jul 29.
An edge-n-coloring of a uniform tiling {\cal T} is uniform if for any two vertices of {\cal T} there is a symmetry of {\cal T} that preserves the colors of the edges and maps one vertex onto the other. This paper gives a method based on group theory and color symmetry theory to arrive at uniform edge-n-colorings of uniform tilings. The method is applied to give a complete enumeration of uniform edge-n-colorings of the uniform tilings of the Euclidean plane, for which the results point to a total of 114 colorings, n = 1, 2, 3, 4, 5. Examples of uniform edge-n-colorings of tilings in the hyperbolic plane and two-dimensional sphere are also presented.
对于均匀铺砌${\cal T}$，如果存在${\cal T}$的一个对称变换，它保持边的颜色不变且将一个顶点映射到另一个顶点，那么该均匀铺砌${\cal T}$的边$n$染色就是均匀的。本文给出了一种基于群论和颜色对称理论的方法，用于得到均匀铺砌的均匀边$n$染色。该方法被应用于对欧几里得平面均匀铺砌的均匀边$n$染色进行完整枚举，结果表明对于$n = 1, 2, 3, 4, 5$，共有114种染色。还给出了双曲平面和二维球面上铺砌的均匀边$n$染色的例子。
