# A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

## TL;DR

This paper extends the Cantor-Bernstein theorem to infinite graphs, demonstrating that under certain conditions, a graph can be decomposed into many spanning trees, with partial results for finite cases.

## Contribution

It establishes a new theorem for infinite graphs relating packings, coverings, and decompositions into spanning trees, advancing understanding in infinite graph theory.

## Key findings

- Infinite graphs with equal packings and coverings admit a spanning tree decomposition.
- Partial results are obtained for finite numbers of spanning trees.
- The question remains open for finite cardinals, with some weaker results proved.

## Abstract

We show that if a graph admits a packing and a covering both consisting of $\lambda$ many spanning trees, where $\lambda$ is some infinite cardinal, then the graph also admits a decomposition into $\lambda$ many spanning trees. For finite $\lambda$ the analogous question remains open, however, a slightly weaker statement is proved.