Acyclic matchings in graphs of bounded maximum degree

A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $\mathrm{\Delta }$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{{\mathrm{\Delta }}^2},$ and we explain how to find such an acyclic matching in polynomial time.