Vertex-coloring 2-edge-weighting of graphs

Abstract

A k - edge-weighting w 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 ∼ e w ( 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 u v ∈ 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 (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.