Vertex-coloring 2-edge-weighting of graphs

A k-edge-weightingw of a graph G is an assignment of an integer weight, w(e)@?{1,...,k}, to each edge e. An edge weighting naturally induces a vertex coloring c by defining c(u)=@?"u"~"ew(e) for every u@?V(G). A k-edge-weighting of a graph G is vertex-coloring if the induced coloring c is proper, i.e., c(u) c(v) for any edge uv@?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 (Karonski 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.