# Rainbow spanning trees in properly coloured complete graphs

## TL;DR

This paper investigates the existence of multiple edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, providing improved bounds and confirming conjectures related to their number.

## Contribution

It improves the lower bound on the number of such trees in properly edge-coloured complete graphs, moving closer to longstanding conjectures.

## Key findings

- Every properly edge-coloured $K_n$ contains $oldsymbol{oldsymbol{ ext{Ω}}(n)}$ edge-disjoint rainbow spanning trees.
- Confirmed that the number of such trees grows linearly with $n$.
- Related work shows the existence of isomorphic rainbow spanning trees, supporting the main results.

## Abstract

In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every $K_n$ properly edge-coloured by $n-1$ colours has $n/2$ edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-coloured $K_n$. Improving the previous best known bound, we show that every properly edge-coloured $K_n$ contains $\Omega(n)$ pairwise edge-disjoint rainbow spanning trees. Independently, Pokrovskiy and Sudakov recently proved that every properly edge-coloured $K_n$ contains $\Omega(n)$ isomorphic pairwise edge-disjoint rainbow spanning trees.