An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\text{gc}$. 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.