论文标题
在Cayley Sum图中的子组完美代码上
On subgroup perfect codes in Cayley sum graphs
论文作者
论文摘要
图$γ$中的完美代码$ c $是$γ$的独立顶点，使得$ c $以外的每个顶点都与$ c $的唯一顶点相邻，而总的完美代码$ c $ in $γ$都是$γ$的一组$γ$的顶点，以便每个顶点$γ$ aukic $γ$ aigacent of uniquent to $ c $ c $ c $ c。令$ g $为有限的组，$ x $是$ g $的普通子集。 Cayley Sum Graph $ \ MATHRM {CS}（g，x）$ g $的连接集$ x $是带顶点集$ g $的图形和两个vertices $ g $，而$ g $和$ h $在x $ in x $和$ G \ g \ \ neq h $时仅相邻。在本文中，我们给出了一些给定组的子组的必要条件，即该组的Cayley Sum图中的（总）完美代码。作为应用程序，将某些小组家庭的Cayley总和图分类为（总）完美代码。
A perfect code $C$ in a graph $Γ$ is an independent set of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $Γ$ is a set of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of a given group being a (total) perfect code in a Cayley sum graph of the group. As applications, the Cayley sum graphs of some families of groups which admit a subgroup as a (total) perfect code are classified.
