Symmetric colorings of G × Z_2

Jabulani Phakathi (School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa)
Yevhen Zelenyuk (School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa)
Yuliya Zelenyuk (School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 2 (2023): Electronic Journal of Graph Theory and Applications

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).