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.
Categories: math.CO