Packing internally disjoint Steiner paths of data center networks

Let S⊆ V(G) and π_G(S) denote the maximum number t of edge-disjoint paths P_1,P_2,…,P_t in a graph G such that V(P_i)∩ V(P_j)=S for any i,j∈{1,2,…,t} and i≠ j. If S=V(G), then π_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 π_G(S)≥ r is NP-complete for a given S⊆ V(G). For an integer r with 2≤ r≤ n, the r-path connectivity of a graph G is defined as π_r(G)=min{π_G(S)|S⊆ 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 significate role in the cloud computing, and prove that π_3(D_k,n)=⌊2n+3k/4⌋ with k≥ 1 and n≥ 6.