  We show that each set of $n\ge 2$ points in the plane in general position has
a straight-line matching with at least $(5n+1)/27$ edges whose segments form a
connected set, and such a matching can be computed in $O(n \log n)$ time. As an
upper bound, we show that for some planar point sets in general position the
largest matching whose segments form a connected set has $\lceil
\frac{n-1}{3}\rceil$ edges. We also consider a colored version, where each edge
of the matching should connect points with different colors.
