The 2-balanced double tree decomposition conjecture

A double tree is a graph formed by taking two trees on the same vertex set, with the two edge sets regarded as distinct even when they join the same pair of vertices. A double tree decomposition partitions the edges of a double tree into two trees. Such a decomposition is $k$-balanced if, at every vertex, the absolute difference between its degrees in the two trees is at most $k$.

2-balanced double tree decomposition conjecture. Any double tree has a $2$-balanced double tree decomposition.

The paper proves that every double tree has a decomposition that is $4$-balanced, improving the previously known bound of five. Examples are known that admit a $2$-balanced decomposition but no $1$-balanced one, so the conjectured bound would be best possible; the conjecture remains open.