Let \(S\subseteq V(G)\) and \(\pi _{G}(S)\) denote the maximum number t of edge-disjoint paths \(P_{1},P_{2},\ldots ,P_{t}\) in a graph G such that \(V(P_{i})\cap V(P_{j})=S\) for any \(i,j\in \{1,2,\ldots ,t\}\) and \(i\ne j\) . If \(S=V(G)\) , then \(\pi _{G}(S)\) is the maximum number of edge-disjoint spanning paths in G. It is proved [Graphs Combin, 37 (2021) 2521–2533] that deciding whether \(\pi _G(S)\ge r\) is NP-complete for a given \(S\subseteq V(G)\) . For an integer r with \(2\le r\le n\) , the r-path connectivity of a graph G is defined as \(\pi _{r}(G)=\) min \(\{\pi _{G}(S)|S\subseteq V(G)\) and \(|S|=r\}\) , which is a generalization of tree connectivity. In this paper, we study the 3-path connectivity of the k-dimensional data center network with n-port switches \(D_{k,n}\) which has signification role in the cloud computing, and prove that \(\pi _{3}(D_{k,n})=\lfloor \frac{2n+3k}{4}\rfloor\) with \(k\ge 0\) and \(n\ge 3\) .
