Covering complete geometric graphs by monotone paths

Given a set A of n points (vertices) in general position in the plane, the complete geometric graph Kn[A] consists of all (n2) 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(n3∕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 randomly selected points, uniformly distributed in [0,1]2, with probability tending to 1 as n→∞, the edge set of Kn[A] can be covered by O(nlogn) crossing-free paths and by O(n√logn) crossing-free matchings. On the other hand, we construct n-element point sets such that covering the edge set of Kn[A] requires a quadratic number of monotone paths.