Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

翻译：给定平面上处于一般位置的 $n$ 个点（顶点）构成的集合 $A$，其\emph{完全几何图} $K_n[A]$ 由 $A$ 中所有 $\binom{n}{2}$ 条线段（边）组成。已知任意 $n$ 个顶点上的完全几何图的边集可被划分为 $O(n^{3/2})$ 条无交叉路径（或匹配）。我们在点集满足不同附加假设的条件下强化了这一结果。特别地，我们证明对于在 $[0,1]^2$ 上均匀分布且\emph{随机}选取的 $n$ 个点构成的集合 $A$，当 $n\rightarrow\infty$ 时，以趋近于 $1$ 的概率，$K_n[A]$ 的边集可被 $O(n\log n)$ 条无交叉路径及 $O(n\sqrt{\log n})$ 个无交叉匹配覆盖。另一方面，我们构造了需要二次数量单调路径才能覆盖 $K_n[A]$ 边集的 $n$ 元点集。