论文标题
在Cayley图中的子组完美代码上
On subgroup perfect codes in Cayley graphs
论文作者
论文摘要
图中的完美代码$γ=（v，e）$是$ v $的子集$ c $，因此$ c $中的两个顶点都相邻，并且每个顶点$ v \ setminus c $ in $ c $中的一个顶点恰好与一个顶点相邻。如果存在$ g $的Cayley图，则$ g $的一个子组$ h $称为$ g $的子组完美代码，该图将$ h $作为完美的代码。同等地，$ h $是$ g $的一个子组完美代码，如果存在一个属于$ g $的子集$ a $ a $ a $ a $ g $，其中包含身份元素，以至于$（a，h）$都是$ g $的瓷砖，因为$ g $的每个元素都可以独特地表示为$ a $ a $ a $和$ h $ h $ h $的元素。 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.
A perfect code in a graph $Γ= (V, E)$ 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.
