On subgroup perfect codes in Cayley graphs

Abstract

A perfect code in a graph $\Gamma =(V,E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if there exists a Cayley graph of $G$ which admits $H$ as a perfect code. Equivalently, $H$ is a subgroup perfect code of $G$ if there exists an inverse-closed subset $A$ of $G$ containing the identity element such that $(A,H)$ is a tiling of $G$ in the sense that every element of $G$ can be uniquely expressed as the product of an element of $A$ and an element of $H$. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving $2$-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and $2$-groups.

This is the corrected version of our paper (arXiv:2006.11104v1) under the same title published in European Journal of Combinatorics 91 (2021) 103228 (https://doi.org/10.1016/j.ejc.2020.103228)