Subgroup regular sets in Cayley graphs

Abstract

Let $\Gamma$ be a graph with vertex set $V$, and let $a$ and $b$ be nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in $\Gamma$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in $\Ga$ are called perfect codes and total perfect codes in $\Ga$, respectively. A subset $C$ of a group $G$ is said to be an $(a,b)$-regular set of $G$ if there exists a Cayley graph of $G$ which admits $C$ as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group $G$ or any group $G$ of order $4p$ or $pq$ for some primes $p$ and $q$, if a nontrivial subgroup $H$ of $G$ is a $(0,1)$-regular set of $G$, then it must also be an $(a,b)$-regular set of $G$ for any $0⩽a⩽|H|-1$ and $0⩽b⩽|H|$ such that $a$ is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.

9 pages

References in corpus (1)

• Corrigendum to "On subgroup perfect codes in Cayley graphs" [European J. Combin. 91 (2021) 103228]

Cited by in corpus (1)

• On regular sets in Cayley graphs