Barát–Thomassen conjecture on tree edge-decompositions

Let $T$ be a tree, and let $G$ be a simple graph. A $T$-edge-decomposition is a decomposition of the edges of $G$ into isomorphic copies of $T$. The size of $G$ is $|E(G)|$, and $G$ is $k$-edge-connected if every nonempty proper vertex set $A$ satisfies $d_G(A)\ge k$, where $d_G(A)$ is the number of edges with exactly one end in $A$.

Barát–Thomassen conjecture. For every tree $T$, there exists a positive integer $k_T$ such that every $k_T$-edge-connected simple graph whose size is divisible by $|E(T)|$ admits a $T$-edge-decomposition.

This is the fundamental edge-decomposition conjecture attributed to Barát and Thomassen (2006). The paper presents stronger quantitative results for graphs with sufficiently large minimum degree, while the general assertion concerns arbitrary simple graphs and remains the motivating conjecture here.