Edge-fault-tolerant strong Menger edge connectivity on the class of hypercube-like networks

From Menger’s theorem, a graph is k-connected if and only if there are at least k-internally disjoint paths between any two distinct vertices. Therefore, the number of internally disjoint paths between two vertices may be larger than the connectivity. Motivated by this observation, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (resp., edge) connectivity as follows. A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x,y in G, there are min{degG(x),degG(y)}(-edge)-disjoint paths between x and y. A graph G is called m(-edge)-fault-tolerant strongly Menger (edge) connected if G−F remains strongly Menger (edge) connected for an arbitrary set F⊆V(G) (resp., F⊆E(G)) with |F|≤m. A graph G is called m-conditional (edge)-fault-tolerant strongly Menger (edge) connected if G−F remains strongly Menger (edge) connected for an arbitrary set F⊆V(G) (resp., F⊆E(G)), |F|≤m and δ(G−F)≥2. In this paper, we show that all n-dimensional hypercube-like networks are (n−2)-edge-fault-tolerant strongly Menger edge connected and (3n−8)-conditional edge-fault-tolerant strongly Menger edge connected for n≥3 which generalizes the results of Qiao and Yang in 2017. Our results are all optimal with respect to the maximum number of tolerated edge faults.