  Each vertex of an arbitrary simple graph on $n$ vertices chooses $k$ random
incident edges. What is the expected number of edges in the original graph that
connect different connected components of the sampled subgraph? We prove that
the answer is $O(n/k)$, when $k\ge c\log n$, for some large enough $c$. We
conjecture that the same holds for smaller values of $k$, possibly for any
$k\ge 2$. Such a result is best possible for any $k\ge 2$. As an application,
we use this sampling result to obtain a one-way communication protocol with
\emph{private} randomness for finding a spanning forest of a graph in which
each vertex sends only ${O}(\sqrt{n}\log n)$ bits to a referee.
