Regular sets in Cayley graphs

TL;DR

This paper explores the properties of perfect sets in Cayley graphs of finite groups, showing how certain normal subgroups can be characterized as perfect sets with various parameters, extending known concepts like perfect and total perfect codes.

Contribution

It proves that non-trivial normal subgroups that are perfect codes can be viewed as (a,b)-perfect sets with specific parameters in Cayley graphs, generalizing previous results.

Findings

Normal subgroups as (a,b)-perfect sets in Cayley graphs

Extension of perfect code concepts to broader (a,b)-perfect sets

Conditions involving gcd for subgroup perfect codes

Abstract

In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.