On a Greedy 2-Matching Algorithm and Hamilton Cycles in Random Graphs   with Minimum Degree at Least Three
  We describe and analyse a simple greedy algorithm \2G\ that finds a good 2-matching $M$ in the random graph $G=G_{n,cn}^{\d\geq 3}$ when $c\geq 15$. A 2-matching is a spanning subgraph of maximum degree two and $G$ is drawn uniformly from graphs with vertex set $[n]$, $cn$ edges and minimum degree at least three. By good we mean that $M$ has $O(\log n)$ components. We then use this 2-matching to build a Hamilton cycle in $O(n^{1.5+o(1)})$ time \whp. 
Alan Frieze
math.CO