Partition of Complete Geometric Graph into Plane Trees

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $P\subset\mathbb R^2$ be a finite set of $n$ points with no three collinear. The complete geometric graph $K_P$ draws each pair as its straight closed segment. Distinct edges are crossing when their relative interiors meet, adjacent when they share an endpoint, and disjoint when the closed segments do not meet. A plane spanning tree is a connected acyclic graph on all the vertices of $P$ in which no two edge interiors intersect.

Statement recorded in the supplied source.

For every even $n\geq2$ and every such $P$, does there exist an edge partition \[E(K_P)=E(T_1)\mathbin{\dot\cup}\cdots\mathbin{\dot\cup}E(T_{n/2})\] into $n/2$ plane spanning trees? [S2]

Short English statement

Can the edges joining an even number of points in general position always be partitioned into non-crossing spanning trees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3084 (OPG-37086). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden contributors, Partition of Complete Geometric Graph into Plane Trees, Open Problem Garden, node 37086. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees.