Perfect codes in vertex-transitive graphs

TL;DR

This paper explores perfect and total perfect codes in vertex-transitive graphs, generalizing subgroup codes from finite groups, providing conditions for their existence, and constructing examples to advance understanding in algebraic graph theory.

Contribution

It introduces a generalized framework for subgroup perfect codes in vertex-transitive graphs, extending previous group-based concepts and establishing necessary and sufficient conditions for their existence.

Findings

Derived a necessary and sufficient condition for subgroup perfect codes in vertex-transitive graphs.

Generalized known results of subgroup perfect codes of finite groups.

Constructed examples of subgroup perfect codes and proposed open problems.

Abstract

Given a graph $\Gamma$, a perfect code in $\Gamma$ is an independent set $C$ of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $\Gamma$ is a set $C$ of vertices of $\Gamma$ such that every vertex of $\Gamma$ is adjacent to a unique vertex in $C$. To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in \cite{HXZ18} as follows: Given a finite group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called a subgroup (total) perfect code of the pair $(G,H)$ if there exists a coset graph $Cos(G,H,U)$ such that the set consisting of left cosets of $H$ in $A$ is a (total) perfect code in $Cos(G,H,U)$. We give a necessary and sufficient condition for a subgroup $A$ of $G$ containing $H$ to be a (total) perfect code of the pair $(G,H)$ and generalize a few known results of subgroup (total) perfect codes of groups. We also construct some examples of subgroup perfect codes of the pair $(G,H)$ and propose a few problems for further research.