Minimum-degree refinement of the Barát–Thomassen conjecture

Let $T$ be a fixed tree with maximum degree $\Delta_T$, and let $|E(T)|$ denote its number of edges. A graph is $T$-decomposable if its edges can be partitioned into sets each inducing a copy of $T$. Minimum-degree refinement of the Barát–Thomassen conjecture. There is a function $f$ such that every graph that is $f(\Delta_T)$-edge-connected, whose number of edges is divisible by $|E(T)|$, and whose minimum degree is at least $f(|E(T)|)$, is $T$-decomposable. This refinement seeks to replace the dependence on a large edge-connectivity bound by separate bounds involving the tree's maximum degree and size. The source presents it as a proposed refinement; no resolution is supplied.