Connected Matchings

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)\mathrm{/}27$ 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.