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On the subgroup regular set in Cayley graphs

Asamin Khaefi, Zeinab Akhlaghi, Behrooz Khosravi

Abstract

A subset $C$ of the vertex set of a graph $Γ$ 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 $a$.

On the subgroup regular set in Cayley graphs

Abstract

A subset of the vertex set of a graph is said to be -regular if induces an -regular subgraph and every vertex outside is adjacent to exactly vertices in . In particular, if is an -regular set of some Cayley graph on a finite group , then is called an -regular set of and a -regular set is called a perfect code of . In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if is a normal subgroup of , then is a perfect code of if and only if it is an -regular set of , for each and with . In this paper, we generalize this result and show that a subgroup of is a perfect code of if and only if it is an -regular set of , for each and such that divides .
Paper Structure (2 sections, 9 theorems, 1 equation, 3 figures)

This paper contains 2 sections, 9 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Lemma 2.3

(see 1 and 9) Let $G$ be a group and $H$ a subgroup of $G$. Then the following are equivalent: (a) $H$ is a perfect code of $G$; (b) there exists an inverse-closed right transversal of $H$ in $G$; (c) for each $x \in G$ such that $x^2 \in H$ and $| H| / | H \cap H^x|$ is odd, there exists $y \in Hx$

Figures (3)

  • Figure 1: (a)$\Gamma_x$(b)$\Delta_x$
  • Figure 2: (a)$\Gamma_x$(b)$\Delta_x$
  • Figure 3: A maximal matching in a complete graph with odd vertices

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Example 2.7
  • ...and 12 more