On Uniform Edge-n-colorings of Tilings

• Main Authors: Abila, Agatha Kristel, De las Penas, Ma. Louise Antonette, Tomenes, Mark
• Format: text
• Published: Archīum Ateneo 2024
• Subjects: uniform edge-n-colorings; uniform tilings; edge orbits; color symmetry; Applied Mathematics; Geometry and Topology; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/284; https://doi.org/10.1107/S2053273324005643
• Institution: Ateneo De Manila University

An edge-n-coloring of a uniform tiling is uniform if for any two vertices of there is a symmetry of 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.