Partitions of complete twisted graphs into plane spanning trees

TL;DR

This paper characterizes how complete twisted graphs can be partitioned into plane spanning trees, revealing that such partitions involve balanced double stars and establishing the existence of large subgraphs with these properties.

Contribution

It provides a complete characterization of partitions of complete twisted graphs into plane spanning trees and identifies conditions for partitions into isomorphic trees.

Findings

Partitions of $T_{2n}$ into plane spanning trees are characterized.

All isomorphic partitions involve balanced double stars.

Large topological subgraphs admit partitions into plane spanning trees.

Abstract

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.