  A 2-matching of a graph $G$ is a spanning subgraph with maximum degree two.
The size of a 2-matching $U$ is the number of edges in $U$ and this is at least
$n-\k(U)$ where $n$ is the number of vertices of $G$ and $\k$ denotes the
number of components. In this paper, we analyze the performance of a greedy
algorithm \textsc{2greedy} for finding a large 2-matching on a random 3-regular
graph. We prove that with high probability, the algorithm outputs a 2-matching
$U$ with $\k(U) = \tilde{\Theta}\of{n^{1/5}}$.
