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An edge weighting problem of a graph G is an assignment of an integer weightto each edge e. Based on edge weighting problem, several types ofvertex-coloring problems are put forward. A simple observation illuminates thatedge weighting problem has a close relationship with special factors of graphs.In this paper, we obtain several results on the existence of factors with thepre-specified degrees, which generalizes earlier results in [2, 3]. Using theseresults, we investigate edge-weighting problem. In particular, we prove thatevery 4-colorable graph admits a vertex-coloring 4-edge-weighting.

L-factors and adjacent vertex-distinguishing edge-weighting