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We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, among others, the following new results: * • If a graph G = (V, E) contains a subgraph isomorphic to a bounded tree-width graph H = (V H , E H ) where |V Hdoi:10.1145/195058.195179 dblp:conf/stoc/AlonYZ94 fatcat:7escc6ce2nfmtowkqhrauzn3ae