Maximizing the maximum degree in ordered nearest neighbor graphs

For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of n points in Rd, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least log n/(4d). Apart from the 1/(4d) factor, this bound is the best
possible. As for the abstract setting, we show that for every n-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree Ω(√︁ log n/ log log n).