Panconnectivity of nn-dimensional torus networks with faulty vertices and edges

The torus network is one of the most popular interconnection topologies for massively parallel computing systems. In this paper, we mainly consider the pp-panconnectivity of nn-dimensional torus networks with faulty elements (vertices and/or edges). A graph GG is said to be pp-panconnected if for each pair of distinct vertices u,v∈V(G)u,v∈V(G), there exists a (u,v)(u,v)-path of each length ranging from pp to |V(G)|−1|V(G)|−1. A graph GG is mm-fault pp-panconnected if G−FG−F is still pp-panconnected for any F⊆V(G)∪E(G)F⊆V(G)∪E(G) with |F|≤m|F|≤m. By using an introduction argument, we prove that the nn-dimensional torus T2k1+1,2k2+1,…,2kn+1T2k1+1,2k2+1,…,2kn+1 is ∑i=1nki-panconnected and (2n−3)(2n−3)-fault [∑i=1n(ki+1)−1]-panconnected.