Edge-fault-tolerance about the SM-{\lambda} property of hypercube-like networks

TL;DR

This paper determines the exact edge-fault-tolerance of hypercube-like networks regarding the SM-{ extlambda} property, which is crucial for network reliability, for a broad range of minimum degree conditions.

Contribution

It provides the first exact value of $sm_\lambda^r(G)$ for hypercube-like networks for any minimum degree threshold $r$, extending previous results limited to smaller $r$.

Findings

Exact value of $sm_\lambda^r(G)$ is $2^r(n-r)-n$ for hypercube-like networks.

The result applies to all hypercube-like networks with $n \geq 3$ and $1 \leq r \leq n-2$.

Enhances understanding of network robustness under edge failures.

Abstract

The edge-fault-tolerance of networks is of great significance to the design and maintenance of networks. For any pair of vertices $u$ and $v$ of the connected graph $G$, if they are connected by $\min \{ \deg_G(u),\deg_G(v)\}$ edge-disjoint paths, then $G$ is strong Menger edge connected (SM-$\lambda$ for short). The conditional edge-fault-tolerance about the SM-$ \lambda$ property of $G$, written $sm_\lambda^r(G)$, is the maximum value of $m$ such that $G-F$ is still SM-$\lambda$ for any edge subset $F$ with $|F|\leq m$ and $\delta(G-F)\geq r$, where $\delta(G-F)$ is the minimum degree of $G-F$. Previously, most of the exact value for $sm_\lambda^r(G)$ is aimed at some well-known networks when $r\leq 2$, and a few of the lower bounds on some well-known networks for $r\geq 3$. In this paper, we firstly determine the exact value of $sm_\lambda^r(G)$ on class of hypercube-like networks (HL-networks for short, including hypercubes, twisted cubes, crossed cubes etc.) for a general $r$, that is, $sm_\lambda^r(G_n)=2^r(n-r)-n$ for every $G_n\in HL_n$, where $n\geq 3$ and $1\leq r \leq n-2$.