Cayley regression problem of some groups (Q6622107)

By a generalized Cayley graph, we mean the following. Given a finite group $G$, for $S\subseteq G$ and $\alpha\in\Aut(G)$, if they satisfy following conditions: (a) $\alpha^2=\mathrm{id}$, where id is the identity of $\AutG)$; (b) for any $g\in G$, $\alpha(g^{-1})g/\in S$; (c) for $g, h\in G$, if $\alpha(h^{-1})g\in S$, then $\alpha(g^{-1})h\in S$. The graph with vertices $G$ and edges $\{g, h\mid \alpha(g^{-1})h\in S\}$ is denoted by $\mathrm{GC}(G, S,\alpha)$. $S$ is called a generalized Cayley subset and $\mathrm{GC}(G, S,\alpha)$ a generalized Cayley graph of $G$ with respect to the ordered pair $(S,\alpha)$. \textit{X. Yang} et al. [Ars Math. Contemp. 15, No. 2, 407--424 (2018; Zbl 1411.05122)] proposed the concept of Cayley regression to discuss the isomorphism problem between the generalized Cayley graphs and the Cayley graphs. Let $G$ be a finite group. (i) $G$ is called an $m$-$\alpha$-Cayley regression if any generalized Cayley graph of $G$ with valency at most $m$ induced by $\alpha\in\Aut(G)$ is isomorphic to some Cayley graph of $G$. (ii) $G$ is called an $m$-Cayley regression if any generalized Cayley graph of $G$ with valency at most $m$ is isomorphic to some Cayley graph of $G$. In particular, if $m=|G|$, we call $G$ an $\alpha$-Cayley regression or a Cayley regression, respectively. The authors in this paper consider the Cayley regression problem of cyclic groups and dihedral groups. They give sufficient and necessary conditions for the cyclic group $Z_n$ to be a Cayley regression. Moreover, the dihedral group $D_2n$ is a 3-Cayley regression if $n$ is a prime power.