# Four Edge-Independent Spanning Trees

## TL;DR

This paper proves a theorem about 4-edge-connected graphs, showing they contain four spanning trees with edge-disjoint paths to a root, and provides a polynomial-time method to construct such trees.

## Contribution

It introduces an ear-decomposition theorem for 4-edge-connected graphs and demonstrates the existence and construction of four edge-independent spanning trees.

## Key findings

- Existence of four edge-independent spanning trees in 4-edge-connected graphs
- Polynomial-time algorithm for constructing these trees
- Extension of ear-decomposition techniques to spanning tree construction

## Abstract

We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every vertex in $G$, the unique paths back to $r$ in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees.