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

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发表日期：

2024年11月15日

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Zhaoman Huang,
Yayu Yang,
Mingzu Zhang,
Xing Yang

摘要

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