  An edge-cut $R$ of an edge-colored connected graph is called a rainbow-cut if
no two edges in the edge-cut are colored the same. An edge-colored graph is
rainbow disconnected if for any two distinct vertices $u$ and $v$ of the graph,
there exists a $u$-$v$-rainbow-cut separating them. For a connected graph $G$,
the rainbow disconnection number of $G$, denoted by rd$(G)$, is defined as the
smallest number of colors that are needed in order to make $G$ rainbow
disconnected.
  In this paper, we first give some tight upper bounds for rd$(G)$, and
moreover, we completely characterize the graphs which meet the upper bound of
the Nordhaus-Gaddum type results obtained early by us. Secondly, we propose a
conjecture that $\lambda^+(G)\leq \textnormal{rd}(G)\leq \lambda^+(G)+1$, where
$\lambda^+(G)$ is the upper edge-connectivity, and prove the conjecture for
many classes of graphs, to support it. Finally, we give the relationship
between rd$(G)$ of a graph $G$ and the rainbow vertex-disconnection number
rvd$(L(G))$ of the line graph $L(G)$ of $G$.
