A Proof of the Bar\'at-Thomassen Conjecture

TL;DR

This paper proves the Barát-Thomassen conjecture, confirming that highly edge-connected graphs can be decomposed into any given tree, extending previous results limited to paths or small-diameter trees.

Contribution

It provides a complete proof of the Barát-Thomassen conjecture for all trees, generalizing earlier partial results.

Findings

Confirmed the conjecture for all trees

Established decomposition conditions for highly edge-connected graphs

Extended previous results beyond paths and small-diameter trees

Abstract

The Bar\'at-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant $k_T$ such that every $k_T$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this conjecture has only been verified when $T$ is a path or when $T$ has diameter at most 4. Here we prove the full statement of the conjecture.