On subgroup perfect codes and regular sets in generalized Cayley graphs
A subset $C$ of the vertex set of a graph $Γ$ is said to be an $(a,b)$-regular set if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. A $(0,1)$-regular set is called a perfect code. Let $G$ be a finite group and let $α\in\operatorname{Aut}(G)$ be such that $α^2=\mathrm{id}$. Let $ω_α(G)=\{α(g^{-1})g:g\in G\}$. For a subset $S\subseteq G$ satisfying $α(S)=S^{-1}$ and $S\capω_α(G)=\emptyset$, the generalized Cayley graph of $(G,α)$ with respect to $S$ is the graph with vertex set $G$ in which two distinct elements $g,h\in G$ are adjacent if and only if $α(g^{-1})h\in S$. In this paper, we prove two main results. First, let $H$ be an $α$-invariant subgroup of a finite group $G$, and let $P$ be an $α$-invariant Sylow 2-subgroup of $H$. We show that $H$ is a perfect code of $(G,α)$ if and only if $P[H,α]$ is a perfect code of $(G,α)$, where $[H,α]=\langle h^{-1}α(h):h\in H\rangle$. Second, we consider finite groups $G$ possessing a cyclic subgroup $A$ of index 2 that is invariant under an involutory automorphism $α$. We show that $ω_α(G)$ is a subgroup of $A$. Using this result, we give a complete characterization of all pairs $(a,b)$ for which an $α$-invariant subgroup $H$ of $G$ is an $(a,b)$-regular set of $(G,α)$. In particular, our characterization applies to all finite groups possessing a characteristic cyclic subgroup of index 2, for every involutory automorphism $α$.