Symmetric Perfect and Symmetric Semiperfect Colorings of Groups

Autor: Rovin B. Santos, Lilibeth D. Valdez, Ma. Lailani B. Walo
Jazyk: angličtina
Rok vydání: 2023
Předmět:
Zdroj: Symmetry, Vol 15, Iss 7, p 1460 (2023)
Druh dokumentu: article
ISSN: 2073-8994
DOI: 10.3390/sym15071460
Popis: Let G be a group. A k-coloring of G is a surjection λ:G→{1,2,…,k}. Equivalently, a k-coloring λ of G is a partition P={P1,P2,…,Pk} of G into k subsets. If gP=P for all g in G, we say that λ is perfect. If hP=P only for all h∈H≤G such that [G:H]=2, then λ is semiperfect. If there is an element g∈G such that λ(x)=λ(gx−1g) for all x∈G, then λ is said to be symmetric. In this research, we relate the notion of symmetric colorings with perfect and semiperfect colorings. Specifically, we identify which perfect and semiperfect colorings are symmetric in relation to the subgroups of G that contain the squares of elements in G, in H, and in G∖H. We also show examples of colored planar patterns that represent symmetric perfect and symmetric semiperfect colorings of some groups.
Databáze: Directory of Open Access Journals
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