Random point sets in the plane allow covering all edges of the complete geometric graph with O(n log n) crossing-free paths whp, while some configurations require Omega(n^2) monotone paths.
Wood, Partitions of complete geometric graphs into plane trees, Computational Geometry 34(2) (2006), 116--125
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Covering Complete Geometric Graphs by Monotone Paths
Random point sets in the plane allow covering all edges of the complete geometric graph with O(n log n) crossing-free paths whp, while some configurations require Omega(n^2) monotone paths.