  The connectivity and edge connectivity of interconnection network determine
the fault tolerance of the network. An interconnection network is usually
viewed as a connected graph, where vertex corresponds processor and edge
corresponds link between two distinct processors. Given a connected graph $G$
with vertex set $V(G)$ and edge set $E(G)$, if for any two distinct vertices
$u,v\in V(G)$, there exist $\min\{d_G(u),d_G(v)\}$ edge-disjoint paths between
$u$ and $v$, then $G$ is strongly Menger edge connected. Let $m$ be an integer
with $m\geq1$. If $G-F_e$ remains strongly Menger edge connected for any
$F_e\subseteq E(G)$ with $|F_e|\leq m$, then $G$ is $m$-edge-fault-tolerant
strongly Menger edge connected. If $G-F_e$ is strongly Menger edge connected
for any $F_e\subseteq E(G)$ with $|F_e|\leq m$ and $\delta(G-F_e)\geq2$, then
$G$ is $m$-conditional edge-fault-tolerant strongly Menger edge connected. In
this paper, we consider the $n$-dimensional bubble-sort star graph $BS_n$. We
show that $BS_n$ is $(2n-5)$-edge-fault-tolerant strongly Menger edge connected
for $n\geq3$ and $(6n-17)$-conditional edge-fault-tolerant strongly Menger edge
connected for $n\geq4$. Moreover, we give some examples to show that our
results are optimal.
