Menger-type connectivity of line graphs of faulty hypercubes

Abstract

A connected graph G is called strongly Menger edge connected if G has min{deg G (x), deg G (y)} edge-disjoint paths between any two distinct vertices x and y in G. In this paper, we consider two types of strongly Menger edge connectivity of the line graphs of n-dimensional hypercube-like networks with faulty edges, namely the m-edgefault-tolerant and m-conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graph of any n-dimensional hypercube-like network is (2n − 4)edge-fault-tolerant strongly Menger edge connected for n ≥ 3 and (4n − 10)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 4. The two bounds for the maximum number of faulty edges are best possible.