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We describe and analyse a simple greedy algorithm \2G\ that finds a good2-matching $M$ in the random graph $G=G_{n,cn}^{\d\geq 3}$ when $c\geq 15$. A2-matching is a spanning subgraph of maximum degree two and $G$ is drawnuniformly from graphs with vertex set $[n]$, $cn$ edges and minimum degree atleast three. By good we mean that $M$ has $O(\log n)$ components. We then usethis 2-matching to build a Hamilton cycle in $O(n^{1.5+o(1)})$ time \whp.

On a Greedy 2-Matching Algorithm and Hamilton Cycles in Random Graphs with Minimum Degree at Least Three