On uniform edge-n-colorings of tilings.

IF 1.9 4区材料科学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Acta Crystallographica Section A: Foundations and Advances Pub Date : 2024-09-01 Epub Date: 2024-07-29 DOI:10.1107/S2053273324005643

Agatha Kristel Abila, Ma Louise Antonette De Las Peñas, Mark Tomenes

引用次数: 0

Abstract

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.

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关于倾斜的均匀边缘-n-着色。

如果对于{\cal T}的任意两个顶点，{\cal T}存在一个对称性，可以保留边的颜色，并将一个顶点映射到另一个顶点上，那么均匀平铺{\cal T}的边-n-着色就是均匀的。本文给出了一种基于群论和颜色对称理论的方法，以求出均匀倾斜图的均匀边-n-颜色。应用该方法完整地列举了欧几里得平面均匀倾斜的均匀边-n-着色，结果表明共有 114 种着色，n = 1、2、3、4、5。此外，还给出了双曲面和二维球面上的均匀边-n-着色的例子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Crystallographica Section A: Foundations and Advances CHEMISTRY, MULTIDISCIPLINARYCRYSTALLOGRAPH-CRYSTALLOGRAPHY

CiteScore

2.60

自引率

11.10%

发文量

419

期刊介绍： Acta Crystallographica Section A: Foundations and Advances publishes articles reporting advances in the theory and practice of all areas of crystallography in the broadest sense. As well as traditional crystallography, this includes nanocrystals, metacrystals, amorphous materials, quasicrystals, synchrotron and XFEL studies, coherent scattering, diffraction imaging, time-resolved studies and the structure of strain and defects in materials. The journal has two parts, a rapid-publication Advances section and the traditional Foundations section. Articles for the Advances section are of particularly high value and impact. They receive expedited treatment and may be highlighted by an accompanying scientific commentary article and a press release. Further details are given in the November 2013 Editorial. The central themes of the journal are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, periodic, quasiperiodic or amorphous, ideal or real, and, on the other, the theoretical and experimental aspects of the various methods to determine these properties and arrangements.