Menger-Type Connectivity of Line Graphs of Generalized Hypercubes With Faulty Edges.

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 the strongly Menger edge connectivity of the line graphs of generalized |$n$| -dimensional hypercubes with faulty edges, namely the |$m$| -edge-fault-tolerant and |$m$| -conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graphs of all generalized |$n$| -dimensional hypercubes are |$(2n-4)$| -edge-fault-tolerant strongly Menger edge connected for |$n\geq 3$| and |$(4n-10)$| -conditional edge-fault-tolerant strongly Menger edge connected for |$n\geq 4$|⁠. The two bounds for the maximum numbers of faulty edges are best possible. [ABSTRACT FROM AUTHOR]