Menger-type connectivity of line graphs of faulty hypercubes

TL;DR

This paper investigates the robustness of line graphs of hypercube-like networks under faulty edges, establishing optimal bounds for their strong Menger edge connectivity in fault-tolerant scenarios.

Contribution

It introduces new bounds for fault-tolerant strong Menger edge connectivity of line graphs of hypercube-like networks, proving these bounds are optimal.

Findings

Line graphs of hypercube-like networks are $(2n-4)$-edge-fault-tolerant strongly Menger edge connected for $n extgreater=3$.

They are $(4n-10)$-conditional edge-fault-tolerant strongly Menger edge connected for $n extgreater=4$.

The bounds for faulty edges are proven to be the best possible.

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 $m$-edge-fault-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 $n\geq 3$ and $(4n-10)$-conditional edge-fault-tolerant strongly Menger edge connected for $n\geq 4$. The two bounds for the maximum number of faulty edges are best possible.