A graph $G$ is one-to-one $r$-disjoint path coverable ($r$-DPC-able for short) if there is an $r$-DPC between any two vertices of $G$. In this paper $G$ is $r$-DPC-able is not same as $G$ is $(r+1)$-DPC-able.
For any other fundamental graph theoretical terminology, please refer to [12].
DCell uses recursively-defined structure to interconnect servers. Each server connects to different levels of DCell through multiple links. We build high-level DCell recursively form many low-level ones. Due to this structure, DCell uses only mini-switches to scale out instead of using high-end switches to scale up, and it scales doubly exponentially with server vertex degree.
We use $\textit{DCell}_k$ to denote a $k$-dimension DCell ($k \ge 0$), $\textit{DCell}0$ is a complete graph on $n$ vertices ($n \ge 2$). Let $t_0$ denote the number of vertices in a $\textit{DCell}0$, where $t_0 = n$. Let $t_k$ denote the number of vertices in a $\textit{DCell}k$ ($k \ge 1$), where $t_k = t{k-1} \times (t{k-1} + 1)$. The vertex of $\textit{DCell}k$ can be labeled by $[\alpha_k, \alpha{k-1}, \dots, \alpha_i, \dots, \alpha_0]$, where $\alpha_i \in {0, 1, \dots, t{i-1}}, i \in {1, 2, \dots, k}$, and $\alpha_0 \in {0, 1, \dots, t_0 - 1}$. According to the definition of $\textit{DCell}_k$ [4, 23], we provide the recursive definition as Definition 1.
Definition 1. The k-dimensional DCell, DCellk, is defined recursively as follows.
(1) $\mathrm{DCell}_0$ is a complete graph consisting of $n$ vertices labeled with $[0],[1],\dots,[n-1]$.
(2) For any $k \ge 1$, $\mathrm{DCell}k$ is built from $t{k-1} + 1$ disjoint copies $\mathrm{DCell}_{k-1}$, according to the following steps.
(2.1) Let $\mathcal{DCell}{k-1}^0$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}{k-1}$ with 0. Let $\mathcal{DCell}{k-1}^1$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}{k-1}$ with 1. ... . Let $\mathcal{DCell}{k-1}^{t{k-1}}$ denote the graph obtained by prefixing the label of each vertex of one copy of $\mathcal{DCell}{k-1}$ with $t{k-1}$. Clearly, $\mathcal{DCell}{k-1}^0 \cong \mathcal{DCell}{k-1}^1 \cong \cdots \cong \mathcal{DCell}{k-1}^{t{k-1}}$.
(2.2) For any $\alpha_k, \beta_k \in {0, 1, \dots, t_{k-1}}$ and $\alpha_k \ge \beta_k (\text{resp. } \alpha_k < \beta_k)$, connecting the vertex $[\alpha_k, \alpha_{k-1}, \dots, \alpha_i, \dots, \alpha_1, \alpha_0]$ of $\mathcal{DCell}{k-1}^{\alpha_k}$ with the vertex $[\beta_k, \beta{k-1}, \dots, \beta_i, \dots, \beta_1, \beta_0]$ of $\mathcal{DCell}_{k-1}^{\beta_k}$ as follow:
\[
\text{(resp.}
\]
\begin{equation}
\left\{
\begin{aligned}
\alpha_k &= \beta_0 + \sum_{j=1}^{k-1} (\beta_j \times t_{j-1}) \\
\beta_k &= \alpha_0 + \sum_{j=1}^{k-1} (\alpha_j \times t_{j-1}) + 1
\end{aligned}
\right.
\tag{2}
\end{equation}
), where $\alpha_i, \beta_i \in {0, 1, \dots, t_{i-1}}, i \in {1, 2, \dots, k}$, and $\alpha_0, \beta_0 \in {0, 1, \dots, t_0 - 1}$.