# Decompositions into isomorphic rainbow spanning trees

## TL;DR

This paper proves that large complete graphs with optimal edge colourings can be decomposed into isomorphic rainbow spanning trees, confirming longstanding conjectures in graph theory.

## Contribution

It establishes the existence of such decompositions in large graphs, resolving conjectures by Brualdi--Hollingsworth and Constantine.

## Key findings

- Decomposition of large complete graphs into isomorphic rainbow spanning trees is always possible.
- Confirms conjectures for sufficiently large graphs.
- Provides a new structural understanding of rainbow decompositions.

## Abstract

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph $K_{2n}$, there exists a decomposition of $K_{2n}$ into isomorphic rainbow spanning trees. This settles conjectures of Brualdi--Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.