We characterize all partitions of the complete twisted graph $T_{ 2n}$ into plane spanning trees. In the case of partitions of $T_{ 2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double stars. As a consequence of our results, any complete topological graph with $n$ vertices contains a complete topological subgraph with $m \geq c\log^{ 1/8} n$ vertices that admits a partition into plane spanning trees.
Categories: math.CO