Vertex-coloring 33-edge-weighting of some graphs

Let GG be a non-trivial graph and k∈Z+k∈Z+. A vertex-coloring kk-edge-weighting is an assignment f:E(G)→{1,…,k}f:E(G)→{1,…,k} such that the induced labeling f:V(G)→Z+f:V(G)→Z+, where f(v)=∑e∈E(v)f(e)f(v)=∑e∈E(v)f(e) is a proper vertex coloring of GG. It is proved in this paper that every  44-edge-connected graph with chromatic number at most  44admits a vertex-coloring  33-edge-weighting.