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In this work we consider the problem of finding Hamilton Cycles in graphs derived from the uniform random intersection graphs model Gn, m, p. In particular, (a) for the case m=nα, α1 we give a result that allows us to apply (with the same probability of success) any algorithm that finds a Hamilton cycle with high probability in a Gn, k graph (i.e. a graph chosen equiprobably form the space of all graphs with k edges), (b) we give an expected polynomial time algorithm for the case p = constant and $m \leq \alpha {\sqrt{{n}\over {{\rm log}n}}}$ for some constant α, and (c) we show that the greedy approach still works well even in the case $m = o({{n}\over{{\rm log}n}})$ and p just above the connectivity threshold of Gn, m, p (found in [21]) by giving a greedy algorithm that finds a Hamilton cycle in those ranges of m, p with high probability.