Reliability analysis for the <i>n</i> th Cartesian product of the balanced complete multipartite graph under various hypotheses

Abstract Analysis of the reliability of networks is crucial to the design and optimization of networks. The $P$-conditional edge-connectivity $\lambda (P;G)$ of a network $G$ is the minimum number of edges whose deletion will divide $G$ into several components, and each component satisfies the property $P$. It proposes a more pinpoint analysis to the reliability of networks. The paper investigates several different the $P$-conditional edge-connectivities of $K_{p;t}^{n}$, such as $d$-embedded edge-connectivity $\lambda (P_{1}^{d};K_{p;t}^{n})$, $(t-1)pd$-good-neighbor edge-connectivity $\lambda (P_{3}^{(t-1)pd};K_{p;t}^{n})$, $(t-1)pd$-average edge-connectivity $\lambda (P_{4}^{(t-1)pd};K_{p;t}^{n})$ for $t\geq 2$, $p\geq 2$ and $0\leq d\leq n-1$, and they possess the same value $(t-1)p(n-d)(tp)^{d}$. Besides, we derive $\lambda (P_{2}^{l}; K_{p;t}^{n})= (t-1)pnl-ex_{l}(K_{p;t}^{n})$ for $1\leq l\leq (tp)^{{\lfloor \frac{n}{2} \rfloor }}$, where $ex_{l}(K_{p;t}^{n})$ denotes the twice of the maximum number of edges in a subgraph induced by $l$ vertices in $K_{p;t}^{n}$. Our method generalizes the result of Yu and Xu in (Comput J 2024; 67: 688–93) and Zhang et al. in (J Supercomput 2022; 78: 7936–47).