Assessing reliability in Complete Josephus Cube networks via strongly Menger edge-connectivity

Abstract

The fault tolerance of an interconnection network relies on its topological parameters of a graph G, with strongly Menger edge-connectivity being a crucial factor. A connected graph G is denoted as strongly Menger edge-connected, if for any two distinct vertices u and v of G, there are $\min \{d_{G}(u), d_{G}(v)\}$ edge-disjoint paths connected u and v, where $d_{G}(u)$ and $d_{G}(v)$ are the degrees of u and v in G. Considering $M\subseteq E(G)$ is a conditional faulty edge set of order t if removing M from the connected graph G ensures the minimum degree of vertices in $G-M$ at least t. Additionally, G is M-strongly Menger edge-connected, if for any $u,v\in V(G-M)$, they are connected by $\min \{d_{G-M}(u), d_{G-M}(v)\}$ edge-disjoint paths in $G-M$. If G is M-strongly Menger edge-connected for any edge subset $M\subseteq E(G)$ satisfying $|M|\le m$, then G is m-fault-tolerant strongly Menger edge-connected. The graph G is m-fault-tolerant strongly Menger edge-connected of order t satisfying that G is m-fault-tolerant strongly Menger edge-connected and $\delta (G-M)\ge t$. The maximum value of m is written as $sm_{\lambda }^{t}(G)$. In this paper, we mainly study the strongly Menger edge-connectedness of the n-dimensional Complete Josephus Cube ($CJC_n$), which is a variant of hypercube. Using the properties of the optimal solution of the edge isoperimetric problem of the $CJC_{n}$, we establish $sm_\lambda ^{t}(CJC_n)=(n-t+2)2^{t-1}-n-2$ to ensure that $CJC_{n}$ maintains m-fault-tolerant strongly Menger edge-connected of order t for two integers $3\le t\le n-2$ and $n \ge 6$. All the results we obtain are optimal in the sense of the maximum number of tolerated edge faults.