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Paper IPM / M / 18134

School of Mathematics
Title:	On the Regular Sets in Cayley Graphs
Author(s):	Zeinab Akhlaghi (Joint with Y. Khaefi and Beh. Khosravi)
Status:	To Appear
Journal:	J. Algebraic Combin.
Supported by:	IPM

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., 2023] 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

$a$ . Also, in [J. Zhang, Y. Zhu, A note on regular sets in Cayley graphs, Bull. Aust. Math. Soc., 2024] it is proved that if

$H$ is a normal subgroup of

$G$ , then

$H$ 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

$a$ and

$b$ is even. We extend this result and we prove that the normality condition is not needed.

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