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In this paper we study the randomly edge colored graph that is obtained byadding randomly colored random edges to an arbitrary randomly edge coloreddense graph. In particular we ask how many colors and how many random edges areneeded so that the resultant graph contains a fixed number of edge disjointrainbow Hamilton cycles. We also ask when in the resultant graph every pair ofvertices is connected by a rainbow path.

How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected?