Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

A weighting of the edges of a graph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph. In this paper we show that such a weighting is possible from the weight set { 1 , 2 , 3 , 4 , 5 } for all graphs not containing components with exactly 2 vertices.