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.

翻译：我们证明了在一般位置下，平面中任意包含$n\ge 2$个点的点集均存在一个直线匹配，其至少包含$(5n+1)/27$条边且这些边对应的线段构成一个连通集，并且此类匹配可在$O(n \log n)$时间内计算得到。作为上界，我们证明了对于某些处于一般位置的平面点集，其边构成连通集的最大匹配恰好包含$\lceil \frac{n-1}{3}\rceil$条边。本文还考虑了着色版本，其中匹配的每条边需连接不同颜色的点。