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Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems;doi:10.1007/s00373-008-0789-5 fatcat:a4j64ccspze2td7cv3f247kf7u