An edge-weighting problem of a graph G is an assignment of an integer
weight to each edge e. Based on an edge-weighting problem, several types of vertexcoloring
problems are put forward. A simple observation illuminates that the edgeweighting
problem has a close relationship with special factors of the graphs. In this
paper, we generalise several earlier results on the existence of factors with pre-specified
degrees and hence investigate the edge-weighting problem — and in particular, we
prove that every 4-colorable graph admits a vertex-coloring 4-edge-weighting.