Strong Menger connectedness of augmented $k$-ary $n$-cubes

TL;DR

This paper investigates the strong Menger connectivity of augmented $k$-ary $n$-cubes, demonstrating their robustness against vertex and edge faults, with optimal fault tolerance bounds established.

Contribution

It establishes the strong Menger (edge) connectivity of augmented $k$-ary $n$-cubes under high fault conditions, extending known properties of these network topologies.

Findings

$AQ_{n,3}$ for $n extgreater 3$ is strongly Menger connected with up to $4n-9$ faulty vertices.

$AQ_{n,k}$ for $n extgreater 1$, $k extgreater 3$ remains strongly Menger connected with up to $4n-8$ faulty vertices.

$AQ_{n,k}$ is strongly Menger edge connected with up to $4n-4$ faulty edges, even under vertex degree restrictions.

Abstract

A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{{\rm deg}_G(x), {\rm deg}_G(y)\}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ for $n\geq 4$ (resp.\ $AQ_{n,k}$ for $n\geq 2$ and $k\geq 4$) is still strongly Menger connected even when there are $4n-9$ (resp.\ $4n-8$) faulty vertices and $AQ_{n,k}$ is still strongly Menger edge connected even when there are $4n-4$ faulty edges for $n\geq 2$ and $k\geq 3$. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that $AQ_{n,k}$ is still strongly Menger edge connected even when there are $8n-10$ faulty edges for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp.\ edge) faults.