# graph theory – Planar pointsets with many heaviest covering cycles

Does there exist for every $$kinmathbb{N}$$ a finite set of points in the euclidean plane such that the heaviest vertex-disjoint cycle cover consists of exactly $$k$$ cycles, when the weight of the cover is defined to be the sum of distances between points that are adjacent on a cycle.

If not, is there an upper bound on $$k$$ and what is it?