A note on harmonic subgraphs in labelled geometric graphs

Let S be a set of n points in general position in the plane, labelled bijectively with the integers {0,1,…,n−1}. Each edge (the straight segment that joins two points) is labelled with the sum of the labels of its endpoints. In this note we investigate the maximum size of noncrossing matchings and paths on S, under the requirement that no two edges have the same weight.