  Let $G$ be a graph and $\mathcal {S}$ be a subset of $Z$. A vertex-coloring
$\mathcal {S}$-edge-weighting of $G$ is an assignment of weight $s$ by the
elements of $\mathcal {S}$ to each edge of $G$ so that adjacent vertices have
different sums of incident edges weights.
  It was proved that every 3-connected bipartite graph admits a vertex-coloring
$\{1,2\}$-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper,
we show that the following result: if a 3-edge-connected bipartite graph $G$
with minimum degree $\delta$ contains a vertex $u\in V(G)$ such that
$d_G(u)=\delta$ and $G-u$ is connected, then $G$ admits a vertex-coloring
$\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. In
particular, we show that every 2-connected and 3-edge-connected bipartite graph
admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in
\{\{0,1\},\{1,2\}\}$. The bound is sharp, since there exists a family of
infinite bipartite graphs which are 2-connected and do not admit
vertex-coloring $\{1,2\}$-edge-weightings or vertex-coloring
$\{0,1\}$-edge-weightings.
