Consider the following open problem: does every complete geometric graph K2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most $n-\sqrt{n/12}$ plane trees.
['Prosenjit Bose', 'Ferran Hurtado', 'Eduardo Rivera-Campo', 'David R. Wood']
Partitions of complete geometric graphs into plane trees