Decomposing highly edge-connected graphs into paths of any given length

TL;DR

This paper proves a longstanding conjecture that highly edge-connected graphs can be decomposed into paths of any fixed length, extending previous results to all path lengths.

Contribution

It confirms the conjecture for paths of any fixed length, broadening the scope of graph decomposition into trees.

Findings

Confirmed the conjecture for all path lengths

Extended previous results to a general class of paths

Provided new techniques for graph decomposition

Abstract

In 2006, Bar\'at and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies of $T$. This conjecture was verified for stars, some bistars, paths of length $3$, $5$, and $2^r$ for every positive integer $r$. We prove that this conjecture holds for paths of any fixed length.