The k-dimensional data center network with n-port switches, denoted by $D_{k,n}$, has been proposed for data centers as a server centric network structure. Wang et al. had proved that $D_{k,n}$ is Hamiltonian-connected for $k\ge 0$ and $n\ge 2$ except for $(k,n)=(1,2)$. In this paper, we prove that for any edge $(u,v)$ in a complete subgraph $K_n$ of $D_{k,n}$, there is a cycle of every length from 3 to $|V(D_{k,n})|$ passing through the edge $(u,v)$ in $D_{k,n}$ for $n\ge 5$. As corollaries, we obtained that $D_{k,n}$ is vertex-pancyclic and pancyclic.

中文翻译：

---

数据中心网络的顶点泛周期性

具有n端口交换机的k维数据中心网络，表示为$d_{ķ，ñ}$已经针对数据中心提出了以服务器为中心的网络结构。Wang等。已经证明$d_{ķ，ñ}$ 是汉密尔顿连接的 $ķ\ge 0$ 和 $ñ\ge 2$ 除了 $（ķ，ñ）=（\mathrm{1个}，2）$。在本文中，我们证明了对于任何优势$（ü，v）$ 在完整的子图中 ${ķ}_{ñ}$ 的 $d_{ķ，ñ}$，每个周期的长度从3到 $|V（d_{ķ，ñ}）|$ 穿过边缘 $（ü，v）$ 在 $d_{ķ，ñ}$ 对于 $ñ\ge 5$。作为推论，我们获得了$d_{ķ，ñ}$ 是顶点全循环和全循环的。