  A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer
weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally
induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every
$u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if
the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in
E(G)$.
  Given a graph $G$ and a vertex coloring $c_0$, does there exist an
edge-weighting such that the induced vertex coloring is $c_0$? We investigate
this problem by considering edge-weightings defined on an abelian group.
  It was proved that every 3-colorable graph admits a vertex-coloring
$3$-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite
graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple
sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In
particular, we show that 3-connected bipartite graphs admit vertex-coloring
2-edge-weighting.
