Symmetric colorings of G × Z_2 - Jabulani Phakathi; Yevhen Zelenyuk; Yuliya Zelenyuk (Electronic Journal of Graph Theory and Applications (EJGTA), 2023)
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).