Edge Partitions of Complete Geometric Graphs (Part 1)

In this note, we disprove the long-standing conjecture that any complete geometric graph on $2n$ vertices can be partitioned into $n$ plane spanning trees. Despite several approaches this conjecture remained open to date. We provide a family of counterexamples based on so-called bumpy wheel sets. To this end, we will give two proofs --- one that is computer assisted (based on an ILP formulation) and a pure pen and paper proof.