Research ArticleNo Access

Counting symmetric colorings of $G \times ℤ_{2}$

Jabulani Phakathi

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050 Johannesburg, South Africa

E-mail Address: Jabulani.Phakathi@wits.ac.za

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Shivani Singh

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050 Johannesburg, South Africa

E-mail Address: 1225827@students.wits.ac.za

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Yevhen Zelenyuk

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050 Johannesburg, South Africa

E-mail Address: yevhen.zelenyuk@wits.ac.za

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, and

Yuliya Zelenyuk

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, 2050 Johannesburg, South Africa

E-mail Address: yuliya.zelenyuk@wits.ac.za

Corresponding author.

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https://doi.org/10.1142/S0219498819501998Cited by:0 (Source: Crossref)

Abstract

Let $G$ be a finite group and let $r \in ℕ$ . An $r$ -coloring of $G$ is any mapping $χ : G \to {1, \dots, r}$ . A coloring $χ$ is symmetric if there is $g \in G$ such that $χ (g x^{- 1} g) = χ (x)$ for every $x \in G$ . We show that if $G$ is Abelian and $f (r)$ is the polynomial representing the number of symmetric $r$ -colorings of $G$ , then the number of symmetric $r$ -colorings of $G \times ℤ_{2}$ is $f (r^{2})$ . We also extend this result to the dihedral group $D (G)$ .

Communicated by T. H. Ha

Keywords:

AMSC: 05A15, 20K01, 05C15, 20D60

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