  We describe an algorithm for finding Hamilton cycles in random graphs. Our
model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from
graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We
focus on the case where $m=cn$ for constant $c$. If $c$ is sufficiently large
then our algorithm runs in $O(n^{1+o(1)})$ time and succeeds w.h.p.
