On the subgroup regular set in Cayley graphs

Abstract: A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set of some Cayley graph on a finite group $G$, then $C$ is called an $(a,b)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if $H$ is a normal subgroup of $G$, then $H$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ with $\gcd(2,|H|-1)\mid a$. In this paper, we generalize this result and show that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ such that $\gcd(2,|H|-1)$ divides $a$.