Vertex-coloring 3-edge-weighting of some graphs

Abstract Let G be a non-trivial graph and k ∈ Z + . A vertex-coloring k -edge-weighting is an assignment f : E ( G ) → { 1 , … , k } such that the induced labeling f : V ( G ) → Z + , where f ( v ) = ∑ e ∈ E ( v ) f ( e ) is a proper vertex coloring of G . It is proved in this paper that every 4 -edge-connected graph with chromatic number at most 4 admits a vertex-coloring 3 -edge-weighting.