Abstract: In this work we consider the problem of finding HamiltonCycles in graphs derived from the uniform random intersection graphs model $G_{n, m, p}$. In particular, (a) for the case $m = n^{\alpha}, \alpha>1$ 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 $G_{n, k}$ graph (i.e. a graph chosen equiprobably form the space of all graphs with $k$ edges), (b) we give an \textbf{expected polynomial time} algorithm for the case $p = \textrm{constant}$ and $m \leq \alpha \sqrt{\frac{n}{\log{n}}}$ for some constant $\alpha$, and (c) we show that the greedy approach still works well even in the case $m = o(\frac{n}{\log{n}})$ and $p$ just above the connectivity threshold of $G_{n, m, p}$ (found in \cite{Singerphd}) by giving a greedy algorithm that finds a Hamilton cycle in those ranges of $m, p$ with high probability.