Symmetric colorings of G × Z_2

Jabulani Phakathi, Yevhen Zelenyuk, Yuliya Zelenyuk

Abstract


Let G be a finite group and let r ∈ N. An r-coloring of G is any mapping χ : G → {1, …, r}. A coloring χ is symmetric if there is g ∈ G such that χ(gx−1g)=χ(x) for every x ∈ G. We show that if f(r) is the polynomial representing the number of symmetric r-colorings of G, then the number of symmetric r-colorings of G × Z2 is f(r2).


Keywords


finite group, symmetric coloring, equivalent colorings, Möbius function, optimal partition

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DOI: http://dx.doi.org/10.5614/ejgta.2023.11.2.3

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