On subgroup perfect codes in vertex-transitive graphs

TL;DR

This paper characterizes subgroup perfect codes in vertex-transitive graphs, explores conditions for their existence, provides counterexamples to a recent question, and investigates maximal subgroups of symmetric groups as perfect codes.

Contribution

It offers a new characterization of subgroup perfect codes in vertex-transitive graphs and constructs counterexamples to a recent open question.

Findings

Characterization of perfect codes in subgroup pairs under certain conditions

Construction of infinitely many counterexamples to a recent conjecture

Initial study of maximal subgroups of symmetric groups as perfect codes

Abstract

A subset $C$ of the vertex set $V$ of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. Given a group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called a perfect code of the pair $(G,H)$ if there exists a coset graph $\mathrm{Cos}(G,H,U)$ such that the set of left cosets of $H$ in $A$ is a perfect code in $\mathrm{Cos}(G,H,U)$. In particular, $A$ is called a perfect code of $G$ if $A$ is a perfect code of the pair $(G,1)$. In this paper, we give a characterization of $A$ to be a perfect code of the pair $(G,H)$ under the assumption that $H$ is a perfect code of $G$. As a corollary, we derive an additional sufficient and necessary condition for $A$ to be a perfect code of $G$. Moreover, we establish conditions under which $A$ is not a perfect code of $(G,H)$, which is applied to construct infinitely many counterexamples to a question posed by Wang and Zhang [\emph{J.~Combin.~Theory~Ser.~A}, 196 (2023) 105737]. Furthermore, we initiate the study of determining which maximal subgroups of $S_n$ are perfect codes.