$[6, 1, 1], [6, 1, 0], [6, 0, 0], [6, 0, 1], [6, 2, 0], [4, 2, 1], [4, 2, 0], [4, 0, 1], [4, 0, 0], [0, 1, 1] >$
$S_1 = <[0, 0, 0], [0, 0, 1], [0, 2, 0]>$,
$S_2 = <[0, 0, 0], [0, 1, 0], [0, 1, 1], [0, 2, 1], [0, 2, 0]>,$
$S_3 = <[0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], [3, 0, 0], [3, 1, 0], [3, 1, 1], [4, 1, 1], [4, 1, 0], [4, 0, 0], [4, 0, 1], [4, 2, 0], [4, 2, 1], [6, 2, 0], [6, 0, 1], [6, 0, 0], [6, 1, 0], [2, 2, 1], [2, 1, 1], [2, 1, 0], [2, 0, 0], [2, 0, 1], [2, 2, 0], [5, 1, 0], [5, 1, 1], [3, 2, 0], [3, 2, 1], [6, 1, 1], [6, 2, 1], [5, 2, 1], [5, 2, 0], [5, 0, 1], [5, 0, 0], [0, 2, 0]>.
$U_1 = <[0, 0, 0], [0, 0, 1], [0, 2, 0], [0, 2, 1]>,$
$U_2 = <[0, 0, 0], [0, 1, 0], [0, 1, 1], [0, 2, 1]>,$
$U_3 = <[0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 2, 0], [1, 2, 1], [1, 1, 1], [1, 1, 0], [3, 0, 1], [3, 0, 0], [3, 1, 0], [2, 1, 0], [2, 0, 0], [2, 0, 1], [2, 2, 0], [5, 1, 0], [5, 0, 0], [5, 0, 1], [5, 2, 0], [4, 2, 0], [4, 0, 1], [4, 0, 0], [4, 1, 0], [2, 1, 1], [2, 2, 1], [6, 1, 0], [6, 1, 1], [6, 2, 1], [5, 2, 1], [5, 1, 1], [3, 2, 0], [3, 2, 1], [3, 1, 1], [4, 1, 1], [4, 2, 1], [6, 2, 0], [6, 0, 1], [6, 0, 0], [0, 2, 1]>$.
□
Lemma 4. For any $\alpha,\beta \in {0,1,\cdots,t_k}$ , $m \in {1,2,\cdots,t_k-3}$ , and $\alpha \neq \beta$ , let $x \in V(DCell_k^\alpha)$ be an arbitrary white vertex , $y \in V(DCell_k^\beta)$ be an arbitrary black vertex , and $G_0 = DCell_k^\alpha \cup DCell_k^\beta \cup (\bigcup_{\theta=0}^m DCell_k^{\omega_\theta})$, where $DCell_k^\alpha$, $DCell_k^\beta$, $DCell_k^{\omega_0}$ , ..., $DCell_k^{\omega_i}$ , ..., $DCell_k^{\omega_m}$ are internally vertex-independent with $i \in {0,1,\cdots,m}$ and $\omega_i \in {0,1,\cdots,t_k}$. Then there exists a path between $x$ and $y$ that containing every vertex in $DCell_k[V(G_0)]$ where $k \geq 1$ and $t_0 = 2$.
Proof. Let $G_1 = DCell_k^\alpha \cup DCell_k^\beta$. Select $z \in V(DCell_k^\alpha)$ and $u \in V(DCell_k^\gamma)$, such that $z \neq x$, $(u,z) \in E(DCell_k)$, and $DCell_k^\gamma \subseteq G_0$, where two graphs $G_1$ and $DCell_k^\gamma$ are internally vertex-independent. Select $\omega \in V(DCell_k^\beta)$ and $v \in V(DCell_k^\delta)$, such that $\omega \neq y$, $(\omega,v) \in E(DCell_k)$, and $DCell_k^\delta \subseteq G_0$ where three graphs $G_1$, $DCell_k^\gamma$, and $DCell_k^\delta$ are internally vertex-independent. Ac- cording to Theorem 1, there exists a path $P$ from $x$ to $z$ that containing every vertex in $DCell_k^\alpha$ and a path $Q$ from $\omega$ to $y$ that containing every vertex in $DCell_k^\beta$. Let $G_2 = G_0[V(\bigcup_{\theta=0}^m DCell_k^{\omega_\theta})]$. We can construct a path $S$ from $u$ to $v$ that containing every vertex in $G_2$ which is similar to Theorem 1. Then there exists a path $P + (z,u) + S + (v,\omega) + Q$ between $x$ and $y$ that containing every vertex in $DCell_k[V(G_0)]$ where $k \geq 1$ and $t_0 = 2$. $\square$
Lemma 5. DCellk is (k + 1)-DPC-able with k ≥ 2 and t₀ = 2.
Proof. We will prove this lemma by induction on the dimension k of DCell. By lemma 3, the lemma holds for t₀ = 2 and k = 2. For t₀ = 2, supposing that the lemma holds for k = τ (τ ≥ 2), we will prove that the lemma holds for k = τ + 1.
For any vertex $x,y \in V(DCell_{\tau+1})$ with $x \neq y$. Let $x \in V(DCell_{\tau}^{\alpha})$ and $y \in V(DCell_{\tau}^{\beta})$ with $\alpha,\beta \in {0,1,\dots,t_{\tau}}$. We can identify $\alpha$ and $\beta$ as follows.
Case 1. $\alpha = \beta$. There exist $(\tau + 1)$ vertex disjoint paths ${P_i|1 \le i \le \tau + 1}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau}^{\alpha}$. Select $u \in V(DCell_{\tau}^{\gamma})$ and $v \in V(DCell_{\tau}^{\delta})$, such that $(x,u), (y,v) \in E(DCell_{\tau+1})$, where three graphs $DCell_{\tau}^{\alpha}$,$DCell_{\tau}^{\gamma}$,$$DCell_{\tau}^{\delta}$ are internally vertex-independent. According to Lemma 4, there exists a path $P_{\tau+2}$ from $u$ to $v$ that visits every vertex in $DCell_{\tau+1}[V(DCell_{\tau+1}-DCell_{\tau}^{\alpha})]$. Then there exist $(\tau+2)$ vertex disjoint paths ${P_i|1 \le i \le \tau + 2}$ between any two distinct vertices $x$ and $y$ of $DCell_{\tau+1}$.