On subgroup perfect codes in vertex-transitive graphs
Binzhou Xia, Junyang Zhang, Zhishuo Zhang
TL;DR
The paper develops a robust characterization of subgroup perfect codes in the context of vertex-transitive graphs by focusing on pairs $(G,H)$ with $H$ a perfect code of $G$. It proves equivalences that connect the existence of inverse-closed left transversals with commutation relations $XH=HX$ and isometric conditions within $A\{g,g^{-1}\}A$, yielding new corollaries and a nonexistence criterion. The authors use these tools to produce infinite counterexamples to a conjectured converse, demonstrating that the converse of a known sufficiency does not hold. They further initiate the study of perfect codes among maximal subgroups of $S_n$, giving complete results for intransitive subgroups and partial results for affine subgroups, supported by computational evidence for small $n$ and suggesting a broad prevalence of maximal subgroups that are perfect codes. Overall, the work bridges group-theoretic structure and graph-theoretic coding concepts, paving the way for a systematic classification of subgroup perfect codes in symmetric groups and related families.
Abstract
A subset $C$ of the vertex set $V$ of a graph $Γ$ is called a perfect code in $Γ$ 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.