Vertex-coloring 2-edge-weighting of graphs

A kk-edge-weighting  ww of a graph GG is an assignment of an integer weight, w(e)∈{1,…,k}w(e)∈{1,…,k}, to each edge ee. An edge weighting naturally induces a vertex coloring cc by defining c(u)=∑u∼ew(e)c(u)=∑u∼ew(e) for every u∈V(G)u∈V(G). A kk-edge-weighting of a graph GG is vertex-coloring   if the induced coloring cc is proper, i.e., c(u)≠c(v)c(u)≠c(v) for any edge uv∈E(G)uv∈E(G).Given a graph GG and a vertex coloring c0c0, does there exist an edge-weighting such that the induced vertex coloring is c0c0? 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 (Karoński et al. (2004) [12]). 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.