Subgroup regular sets in Cayley graphs

TL;DR

This paper investigates the structure of regular sets in Cayley graphs of specific groups, establishing conditions under which certain subgroups are regular sets with various parameters, extending known concepts like perfect codes.

Contribution

It proves that for certain groups, nontrivial subgroups that are (0,1)-regular sets are also (a,b)-regular sets for a range of parameters, generalizing the concept of perfect codes.

Findings

Subgroups as (0,1)-regular sets imply (a,b)-regularity for specific parameters.

Results apply to generalized dihedral groups and groups of order 4p or pq.

Conditions involve parity of a when subgroup order is odd.

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 $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ 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.