On Conditional Edge-Fault-Tolerant Strong Menger Edge Connectivity Of Folded Hypercubes

Abstract

Abstract Edge connectivity is an important parameter for the reliability of the inter-connection network. A graph $G$ is strong Menger edge-connected ($SM$-$\lambda $ for short) if there exist min$\{\deg _{G}(u),\deg _{G}(v)\}$ edge-disjoint paths between any pair of vertices $u$ and $v$ of $G$. The conditional edge-fault-tolerance strong Menger edge connectivity of $G$, denoted by $sm_{\lambda }^{r}(G)$, is the maximum integer $m$ such that $G-F$ remains $SM$-$\lambda $ for any edge set $F$ with $|F|\leq m$ and $\delta (G-F)\geq r$, where $\delta (G-F)\geq r$ is the minimum degree of $G-F$. Most of the previous papers discussed $sm_{\lambda }^{r}(G)$ in the case of $r\leq 2$. In this paper, we show that $sm_{\lambda }^{r}(FQ_{n})=2^{r}(n-r+1)-(n+1)$ for $1\leq r\leq n-2$, where $n\geq 4$.