Edge-fault-tolerance about the SM-{\lambda} property of hypercube-like networks

The edge-fault-tolerance of networks is of great significance to the design and maintenance of networks. For any pair of vertices $u$ and $v$ of the connected graph $G$, if they are connected by $\min \{ \deg_G(u),\deg_G(v)\}$ edge-disjoint paths, then $G$ is strong Menger edge connected (SM-$\lambda$ for short). The conditional edge-fault-tolerance about the SM-$ \lambda$ property of $G$, written $sm_\lambda^r(G)$, is the maximum value of $m$ such that $G-F$ is still SM-$\lambda$ for any edge subset $F$ with $|F|\leq m$ and $\delta(G-F)\geq r$, where $\delta(G-F)$ is the minimum degree of $G-F$. Previously, most of the exact value for $sm_\lambda^r(G)$ is aimed at some well-known networks when $r\leq 2$, and a few of the lower bounds on some well-known networks for $r\geq 3$. In this paper, we firstly determine the exact value of $sm_\lambda^r(G)$ on class of hypercube-like networks (HL-networks for short, including hypercubes, twisted cubes, crossed cubes etc.) for a general $r$, that is, $sm_\lambda^r(G_n)=2^r(n-r)-n$ for every $G_n\in HL_n$, where $n\geq 3$ and $1\leq r \leq n-2$.
Comment: 9 pages, 3 figures